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TREATISE 


ON 


ALGEBRA, 


roR  THE  USX  or 


SCHOOLS  AND   COLLEGES, 


^yy-y  : 


WILLIAM    SMYTH,   A.M.     ''^  ^ 

PEOFESSOE    OF    MATHEMATICS    IN    BOWDOIN    COLLEGE. 


BOSTON: 

SANBORN,   CARTER   &   BAZIN, 

PORTLAND: 

BLAKE    &    CARTER. 

1855. 


A.^^ 


Entered  according  to  Act  of  Congress,  in  the  year  1862, 

BY    WILLIAM    SMYTH,  A.M. 

In  th«  Clerk's  Office  of  the  Diatrict  Court  of  the  District  of  Maine. 


Stereotyped  by 

ROBART    k    nOBBINS, 
KNGLAUD  TTPR  AND  STEREOTYrB  FOCNDBBT, 
BOSTON. 


Qk 


^3 


Sicl 


PREFACE 


The  present  Treatise  is  composed  substantially  of  the  Elements  of  Alge' 
bra  of  tlic  author,  -with  such  additional  matter  as  fully  to  adapt  it  to  the 
advanced  course  of  mathematics  now  generally  pursued  in  the  American 
Colleges.  In  its  preparation  the  object  has  been  to  give  a  clear  view  of 
the  nature  and  powers  of  Algebra.  The  analytic  method  is  uniformly 
pursued,  and  the  topics  are  so  presented  as,  in  general,  to  lead  the 
student  to  feel  the  want  of  a  new  principle  before  proceeding  to  its  investi- 
gation. Thus,  the  work  commences  with  the  exposition  of  Algebra,  as  a 
concise  language  adapted  to  facilitate  the  processes  of  reasoning  required 
in  mathematical  investigations.  The  operations  of  Algebra,  therefore,  with 
which  most  treatises  begin,  are  not  introduced  until,  in  the  use  of  the 
algebraic  language  in  the  solution  of  questions,  the  manner  in  which  these 
operations  arise,  and  the  reason  for  them,  are  seen.  The  same  general 
plan  is  pursued  throughout.  Much  attention  is  paid  to  the  Discussion  of 
Problems  and  Equations,  a  topic  of  the  highest  importance  to  the  clear 
understanding  of  the  true  nature  of  Algebra.  A  section  is  given  on  the 
Indeterminate  Analysis,  a  subject  not  usually  introduced  into  our  text 
books,  but  of  great  value  in  itself  and  in  its  relation  to  Analytic  Geometry. 
A  full  view  is  given  of  the  General  Theory  of  Equations,  and  of  the  method 
of  solving  Numerical  Equations  of  any  degree.  The  several  subjects  are 
presented  in  the  manner  found  by  experience  best  adapted  to  the  conven- 
ience of  recitations  and  the  progress  of  the  pupil.  All  needed  help,  it  is 
believed,  is  furnished,  without  that  difFuseness  of  explanation  which  leaves 
to  the  learner  no  room  for  the  exercise  of  his  own  powers.  The  difficulties 
to  be  encountered  are  such  only  as  pertain  of  necessity  to  the  subject,  and 
which  serve  to  furnish  a  healthy  stimulus  to  exertion  and  the  njental 
discipline  necessary  to  the  successful  prosecution  of  more  advanced  studies. 

The  work  in  its  present  form  is  still  better  adapted,  it  is  hoped,  to  the 
use  of  Academies  and  High  Schools,  in  which  it  has  heretofore  been  exten- 
sively used.  For  a  younger  class  of  pupils,  the  Elementaiy  Algebra  of 
the  author  will  be  found  sufficiently  simple,  and  an  easy  introduction  to 
the  present  work.  Wm   Smyth 

Bowdoin  College,  1853 


TABLE  OF  CONTENTS. 


BKCTION  PAGB 

1 .  Explanation  of  Algebraic  Signs 1 — 9 

2.  Equations 9—22 

3.  Algebraic  Operations.     Addition  —  Subtraction  —  Multi- 

plication and  Division  of  Algebraic  Quantities   .    .    .  22 — 44 

4.  Algebraic  Fractions.     Greatest  Common  Divisor  ....         44 — 55 

5.  Equations  of  the  First  Degree.     Elimination 55 — C9 

6.  Negative  Quantities C9 — 7G 

7.  Indeterminate  Analysis      7C — 87 

8.  Solution  of  Questions  in  a  general  manner 87 — 100 

9.  Discussion  of  Problems  and  Equations  of  the  First  Degree  100 — 111 

10.  Theory  of  Inequalities Ill — 115 

11.  Extraction  of  the  Square  Root 115 — 135 

12.  Equations  of  the  Second  Degree 135 — 149 

13.  Discussion  of  Equations  and  Problems  of  the  Second  Degree  149 — 1G5 

14.  Maxima  and  Minima 165 — 1C8 

15.  Powers  and  PkOots  of  Monomials 1G8 — 171 

IG.        Powers  of  Compound  Quantities.     Theory  of  Combinations. 

Binomial  Theorem 171—184 

17.  Roots  of  Compound  Quantities 184—191 

18.  Calculus  of  Radical  Expressions 191 — 197 

19.  Theory  of  Exponents 197—203 

20.  Proportions 203—210 

21.  Progressions 210—226 

22.  Theory  of  Continued  Fractions 226—235 

23.  Exponential  Equations.     Logarithms 235 — 248 

24.  Application  of  Logarithms.     Compound  Interest.     Annui- 

ties       248—257 

25.  Praxis.     Equations  of  the  First  and  Second  Degree   .    .    .  257 — 264 

26.  General    Theory  of    Equations.      Detached    Coefficients. 

Synthetic  Division.  Divisibility  of  Equations.  Num- 
ber of  the  Roots.  Coefficients.  Form  of  the  Roots. 
Signs  of  the  Roots.  Limits  of  the  Roots.  Limiting 
Equation.     Equal  Roots.     Imaginary  and  Real  Roots. 

Sturm's  Theorem 265    -309 

27.  Numerical    Equations  of  any  Degree.      Commensurable 

Roots.     Incommensurable  Roots.     Homer's  Method  of 

Approximation 309   -323 

28.  Elimination  by  means  of  the  Greatest  Common  Divisor  .  323- -326 

29.  Infinite   Scrie?.     Undetermined   Coefficients.     The  Differ- 

ential Method.     Interpolation.     Summation  of  Series  32G— 330 


ELEMENTS  OF  ALGEBRA. 


SECTION  I. — Explanation  of  Algebraic   Signs. 

1.  Let  it  be  proposed  to  divide  the  number  56  into  two  such 
parts,  that  the  greater  may  exceed  the  less  by  12. 

To  resolve  this  question,  we  remark  that, 

1°.   The  greater  part  is  equal  to  the  less  added  to  12. 

2°.   The  greater  party  added  to  the  less  part,  is  equal  to  5Q. 

It  follows,  therefore,  that, 

3°.  The  less  part,  added  to  12,  added  also  to  the  less  part,  is 
iqual  to  5Q. 

But  this  language  may  be  abridged,  thus, 

4°.   Tioice  the  less  part,  added  to  12,  is  equal  to  56 ;  whence, 

5°.   Twice  the  less  part  is  equal  to  56  diminished  by  12. 

Subtracting,  therefore,  12  from  56,  we  have 

6°.   Twice  the  less  part  equal  to  44 ;  wherefore 

7°.  Once  the  less  part  is  equal  to  44  divided  by  2,  or  perform- 
ing the  division,  we  have 

8°.   Once  the  less  part  equal  to  22. 

Adding  12  to  22  we  have  34  for  the  greater  part.  The  parts 
required,  therefore, ^re  22  and  34. 

2.  In  the  process  of  reasoning  required  in  the  solution  of  the 
proposed  question  expressions,  such  as  "  added  to,"  "  diminished 
by,"  ^^  eqical  to,"  &c.  are  often  repeated.      These  expressions 


6  ELE'tfSNTS    OF   ALGEBRA. 

refer  to  the  operations,  by  which  the  numbers  given  in  the  ques- 
tion are  connected  among  themselves,  or  to  the  relations  which 
they  bear  to  each  other.  The  reasoning,  therefore,  which  per- 
tams  to  the  solution  of  the  proposed,  it  is  evident,  may  be  rendered 
much  more  concise,  by  representing  each  of  these  expressions  by 
a  convenient  sign. 

It  is  agreed  among  mathematicians  to  represent  the  expression 
"  added  to'^  by  the  sign  -j-,  read  plus,  the  expression  "  diminished 
}nf^  by  the  sign  — ,  read  minus,  the  expression  "  multiplied  3y" 
by  the  sign  X>  that  of  "  divided  ly"  by  the  sign  -f-.  Lastly,  the 
expression  "  equal  to"  is  represented  by  the  sign  =. 

3.  By  means  of  the  above  signs,  the  reasoning  in  the  question 
proposed  may  be  much  abridged ;  still,  however,  we  have  frequent 
occasion  to  repeat  the  expression  "  the  less  part."  The  reasoning, 
therefore,  may  be  still  more  abridged  by  representing  this  also 
by  a  sign. 

The  less  part  is  the  unknown  quantity  sought  directly  by  the 
*  reasoning  pursued.     It  is  agreed  in  general  to  represent  the 
unknown  quantity  or  quantities  sought  in  a  question  by  some 
one  of  the  last  letters  of  the  alphabet,  as,  x,  y,  z. 

4.  Let  us  now  resume  the  question  proposed,  and  employ  in 
its  solution  the  signs,  which  have  been  explained. 

Let  us  represent  by  x  the  less  of  the  two  parts  required,  we 
have  then 

a:  -f-  12  =  the  greater  part, 
X  +  12 +  x==5Q 
2  X  ^  +  12  =  56 

2  X  a:  =  56  —  12 
2  X  a:  =  44 

x  =  U-^2 
.^^  X  =  22. 

The'  multiplication  of  a;  by  2  may  be  expressed  more  con- 
cisely thus,  2.x,  or  still  more  concisely  thus,  2x.  Division 
also  is  more  commonly  indicated  by  writing  the  number  to  be 
diyided  above  a  horizontal  line,  and  the  divisor  beneath  it  in 


EQUATIONS   OF   THE    FIRST   DEGREE.  7 

the  form  of  a  fraction ;  14  divided  by  2,  for  example,  is  indicated 
thus,-^-. 

5.  The  question,  which  we  have  solved,  is  simple;  it  is 
sufficient,  however,  to  show  the  aid  which  may  be  derived 
from  convenient  signs  in  facilitating  the  reasonings,  that  per- 
tain to  the  solution  of  a  question.  Indeed  in  abstruse  and 
complicated  questions,  it  would  often  be  difficult,  and  sometimes 
absolutely  impossible  to  conduct,  without  such  aid,  the  reasonings 
required. 

6.  The  signs  which  have  been  explained, '  together  with 
those  which  will  hereafter  be  introduced,  are  called  Algebraic 
signs.  It  is  from  the  use  of  these  that  the  science  of  Algebra  is 
derived. 

Let  us  now  employ  the  signs  already  explained  in  the  solution 
of  some  questions. 

1.  Three  men,  A,  B,  and  C  trade  in  company  and  gain  $405, 
of  which  B  has  twice  as  much  as  A,  and  C  three  times  as  much 
as  B.     Required  the  share  of  each. 

Let  X  represent  the  share  of  A,  then  2a;  will  represent  the 

share  of  B  and  6a;  the  share  of  C.     Then,  since  the  shares  added 

together  should  be  equal  to  the  sum  gained,  we  have 

a;_|-2a;  +  6a;  =  405 

9a;  =  405 

405       ._ 
a;^_  =  45. 

Thus  we  have  A's  share  =  $45;  whence  B's  share  is  ^90 
and  C's  $270. 

2.  A  fortress  is  garrisoned  by  2600  men ;  and  there  are  nine 
times  as  many  infantry,  and  three  times  as  many  artillery  as 
cavalry.     How  many  are  there  of  each  ? 

3.  From  two  towns,  which  are  187  miles  distant,  two  travel- 
lers set  out  at  the  same  time,  with  an  intention  of  meeting.  One 
of  them  goes  8  miles,  and  the  other  9  miles  a  day.  In  how 
many  days  will  they  meet  ^ 


m- 


8  ELEMENTS    OF   ALGEBRA. 

4.  A  gentleman  meeting  four  poor  persons  distributed  5  shil- 
lings among  them;  to  the  second  lie  gave  twice,  to  the  third 
thrice,  and  to  the  fourth  four  times  as  much  as  to  the  first.  What 
did  he  give  to  each  ? 

5.  Four  persons,  A,  B,  C  and  D  made  a  joint  stock;  B  puts' 
in  twice  as  much  as  A,  C  puts  in  three  times  as  much  as  B,  and 
D  puts  in  as  much  as  the  other  three  together.  The  whole  stock 
is  $20,000.     How  much  did  each  put  in  ? 

6.  To  divide  the  number  230  into  three  such  parts,  that  the 
excess  of  the  mean  above  the  least  may  be  40,  and  the  excess  of 
the  greatest  above  the  mean  may  be  60. 

Let  X  represent  the  least  part,  then  rr  -[-  40  will  be  the  mean, 
and  a;  -j-  40  -|-  60  will  be  the  greatest  part ;  we  have  therefore 
2; -I- a:  +  40  +  a;  +  40  +  60  =  230 
8a; +140  =  230 
3a:  =90 
a:  =  30. 

The  parts  will  then  be  30,  70  and  130  respectively. 

7.  A  draper  bought  three  pieces  of  cloth  which  together  mea- 
sured 159  yards.  The  second  piece  was  15  yds.  longer  than  the 
first,  and  the  third  24  yds.  longer  than  the  second.  What  was 
the  length  of  each  ? 

8.  Three  men.  A,  B  and  C  made  a  joint  stock ;  A  puts  in  a 
certain  sum,  B  puts  in  SI  15  more  than  A,  and  C  puts  in  $235 
more  that  B ;  the  whole  stock  was  $1753.  What  did  each  man 
put  in  ? 

9.  A  gentleman  buys  4  horses,  for  the  second  of  which  he 
gives  £12  more  than  for  the  first,  for  the  third  £6  more  than  for 
the  second,  and  for  the  fourth  £2  more  than  for  the  third.  The 
sum  paid  for  all  was  £230.     How  much  did  each  cost  ? 

10.  A  man  leaves  by  will  his  property,  amounting  to  $14000, 
to  his  wife,  two  sons  and  three  daughters ;  each  son  is  to  receive 
twice  as  much  as  a  daughter,  and  the  wife  as  much  as  all  the 
children  together.     What  will  each  receive  ? 


,   EQUATIONS   OF   THE    FIRST    DEGREE.  9 

11.  An  express  sets  out  to  travel  240  miles  in  4  days,  but  in 
consequence  of  the  badness  of  the  roads,  he  found  he  must  go  5 
miles  the  second  day,  9  the  third  and  14  the  fourth  day  less  than 
the  first.     How  many  miles  must  he  travel  each  day  ? 

12;  The  sum  of  8300  was  divided  among  4  persons;  the 
second  received  three  times  as  much  as  the  first,  the  third  as 
much  as  the  first  and  second,  and  the  fourth  as  much  as  the 
second  and  third.     What  did  each  receive  ? 

13.  A  silversmith  has  3  pieces  of  metal.  The  second  weighs 
6  oz.  more  than  twice  the  first,  and  the  third  9  oz.  more  than 
three  times  the  second.  The  weight  of  the  whole  being  52  oz., 
what  is  the  weight  of  each  ? 


SECTION  II.— Equation^. 


7.  The  difference  between  two  numbers  is  25  and  the  greater 
is  4  times  the  less ;  required  the  numbers. 

Let  X  represent  the   less,   then  a:-f-25  will  represent  the 

gTeater ;  but  since  by  the  question  the  greater  is  four  times  the 

less,  42;  will  also  represent  the  greater;  these  two  expressions 

for  the  same  thing  will  therefore  be  equal  to  each  other,  and  we 

have 

a:4-25  =  4a:. 

An  expression  for  the  equality  of  two  things  is  called  an 
equation.  The. two  equal  quantities,  of  which  an  equation  is 
composed,  are  called  members  of  the  equation ;  the  one  on  the 
left  of  the  sign  of  equality  is  called  the  first  member  and  the 
other  the  second. 

If  a  member  consists  of  parts  separated  by  the  signs  -[-  and  — , 
these  parts  are  called  terms. 

Thus  in  the  equation  a;  -|-  25  =  4  a:,  the  expression  a:  -|-  25  is 
the  first  member  and  4  a:  the  second. 

The  quantities  x  and  25  are  the  terms  of  the  first  member. 


10  ELEMENTS    OF    ALGEBRA. 

A  figure  written  before  a  letter,  showing  how  many  times  the 
letter  is  to  be  taken,  is  called  the  coefficient  of  that  letter.  In 
the  quantities  4  a:,  7  a;,  4  and  7  are  the  coefficients  of  x. 

Equations  are  distinguished  into  different  degrees.  An  equa* 
tion,  in  which  the  unknown  quantity  is  neither  multiplied  by 
itself,  nor  by  any  other  unknown  quantity,  is  called  an  equation 
of  the  first  degree. 

8.  In  the  solution  of  a  question  by  the  aid  of  algebraic  signs 
there  are,  it  is  evident  from  the  examples  already  performed,  two 
distinct  parts.  In  the  first,  we  form  an  equation  by  means  of  the 
relations  established  by  the  nature  of  the  question  between  the 
known  and  unknown  quantities.  This  is  called  putting  the 
question  into  an  equation. 

In  the  second  part,  from  the  equation,  thus  formed,  we  deduce 
a  series  of  other  equations,  the  last  of  which  gives  the  value 
of  the  unknown  quantity.  This  is  called  resolving  or  reducing 
the  equation. 

9.  No  general  and  exact  rule  can  be  given  for  putting  a  ques- 
tion into  an  equation  When  however  the  equation  of  a  question 
is  formed,  there  are  regular  steps  for  its  reduction,  which  we 
shall  now  explain. 

Since  the  two  members  of  an  equation  are  equal  quantities, 
it  is  evident,  that,  1°.  the  same  quantity  may  be  added  to  both 
sides  of  an  equation  ivithout  destroying  the  equality ;  2°.  the 
same  quantity  may  be  subtracted  from  both  sides  of  an  equation 
without  destroying  the  equality;  3°.  both  sides  of  an  equation 
fnay  be  multiplied,  or  4°.  both  sides  may  be  divided  by  the  same 
^quantity  without  destroying  the  equality. 

10.  Let  it  be  proposed  to  resolve  the  equation  derived  trom 
the  following  enunciation,  viz.  To  find  a  number  such  that  if 
one  half  and  one  third  of  this  number  be  added  to  itself  the  sum 
will  be  equal  to  30. 

Let  X  represent  the  number,  then  one  half  of  this  number  wiD 


EQUATIONS    OF   THE    FIRST   DEGREE.  11 

1  X  1  X 

be  represented  by  ^  a;  or  -  and  one  third  by  ^  a;  or  ^r,  and  we 
have 

.+1+5=30. 

To  resolve  this  equation  we  must  free  the  fractional  terms  from 
their  denominators.  In  order  to  this  we  multiply  both  sides  of 
the  equation  first  by  2,  which  gives 

2a;  +  a:  +  ^|  =  60; 

multiplying  next  by  three,  we  have 

6a;  +  3a;  +  2a:  =180, 
an  equation  free  from  denominators.     To  free  an  equation  there- 
fore from  denominators,  multiply  the  equation  by  the  denomina' 
tors  successively. 

Ex.  1.     Free  from  denominators  the  equation 

-  -4-  -  —  -  —  49 

Ex.  2.     Free  from  denominators  the  equation 

3^7       12^11"" 
Since  in  this  equation  the  denominator  12  is  a  multiple  of  3, 
multiplying  by  12,  we  have 

4.  +  ~-x  +  ^=156. 

Thus  by  multiplying  first  by  12,  the  number  of  multiplications 
necessary  to  free  the  equation  from  denominators  is  diminished, 
and  the  equation  itself,  when  freed  from  denominators,  is  left  m 
*  more  simple  state. 

Ex.  3.     Free  from  denominators  the  equation 

-4-^  —  —  —  —  —  120 
7^9       21       18~"       • 

Ex.  4.     Free  from  denominators  the  equation 


± zj-Jl f_L    in 

6      4"^  12      3'^2~^"' 


12  ELEMENTS    OF   ALGEBRA. 

Ex.  5.     Free  from  denominators  the  equation 

The  least  number  divisible  by  each  one  of  the  denominators 
of  the  proposed,  it  is  easy  to  see,  is  20.  Multiplying  by  20,  we 
have 

10a:  +  2a:  +  15a;  — 4a:+ 120  =  180; 
thus  the  proposed  is  freed  at  once  from  denominators,  and  the 
equation  which  results,  it  is  evident,  is  the  most  simple  to  which 
it  can  be  reduced  free  from  denominators. 

From  what  has  been  done,  we  have  the  following  rule  to  free 
an  equation  from  denominators,  viz.  Find  the  least  common 
multiple  of  the  denominators ;  multiply  each  term  hy  this  com- 
mon multiple,  observing  to  divide,  as  we  proceed,  the  numerator 
of  each  fractional  term  hy  its  denominator. 

11.     Let  it  be  proposed  to  resolve  the  equation 

3a:  +  25  =  60  — 4a;. 
To  resolve  this  equation,  it  will  be  necessary  to  transfer  tho 
terms  25  and  4a;  from  the  members,  in  which  they  now  stand,  to 
the  opposite.     In  order  to  this,  let  us  first  subtract  25  from  both 
members,  we  then  have 

3:^+25  —  25  =  60  — 4a;  — 25. 
or  3a;  =  60  — 4a;  — 25. 

Adding  next  4  a;  to  both  sides  of  this  last,  we  have 
3z  +  4a;  =  60  +  4a;  —  4a;  —  25. 
or  3a;  +  4a;  =  60  — 25. 

Comparing  the  last  equation  with  the  proposed,  the  term  25 
which  is  additive  in  the  first  member  has,  it  is  evident,  passed 
into  the  second  member  with  the  sign  of  subtraction,  and  the 
term  4  a;  which  was  subtractive  in  the  second  member  has  passed 
into  the  first  with  the  sign  of  addition.  Whence  the  following 
rule,  for  transposing  a  term  from  one  member  of  an  equation  to 
the  other,  will  be  readily  inferred,  viz.  Efface  the  term  in  the 
member  in  ivhich  it  stands,  and  write  it  in  the  other  with  the 
contrary  sign. 


EQUATIONS   OF   THE    FIRST   DEGREE.  13 

12.  Let  it  be  proposed  next  to  resolve  the  equation 

5x      4x       y. 7 13a; 

IS  —  y""  8        6"* 

Freeing  from  denominators,  we  have 

lOo;  —  S2x  —  312  =  21  —  52a; ; 

transposing  and  reducing,  we  have 
30  a;  =  333; 

whence  dividing  both  sides  by  30  we  obtain 
a;=llyV- 

The  unknown  quantity  in  equations  of  the  first  degree  can  be 
combined  with  those  which  are  known  in  four  different  ways 
only,  viz.  by  addition,  subtraction,  multiplication,  and  division. 
From  what  has  been  done,  we  have  therefore  the  following  rule 
for  the  resolution  of  equations  of  the  first  degree  with  one  un- 
known quantity,  viz.  1°.  Free  the  proposed  equation  from  de- 
nominators ;  2°.  bring  all  the  terms,  which  contain  the  unknown 
quantity  into  the  first  member  and  all  the  knoion  quantities  into 
the  other  ;  3°.  unite  in  one  term  the  terms  which  contain  the  un- 
known quantity,  and  the  knoion  quantities  in  another  ;  4°.  divide 
both  sides  by  the  coefficient  of  the  unknown  quantity. 

13.  Applying  ^he  above  rule  to  the  equation 

^_.|  +  10  =  |-|+ll,weobtaina;=12. 

In  order  to  verify  this  result  we  substitute  12  for  z  in  the  pro- 
posed, it  then  becomes 

lH_l?4-io-lH_l?+ii- 

6        4+^"— 3        2+^^' 
whence  performing  the  operations  indicated  we  obtain 

9  =  9. 
The  value  a;  =  12  satisfies  therefore  the  proposed  equation. 

In  general,  to  verify  the  value  of  the  unknown  quantity  de- 
duced from  an  equation,  we  substitute  this  value  for  the  unknown 
quantity  in  the  equation.  If  this  renders  the  two  members  iden- 
tically the  same,  the  answer  is  correct. 

B 


14  ELEMENTS   OF   ALGEBRA. 

14.   The  following  examples  will  serve  as  an  exercise  for  the 
learner  in  the  reduction  of  equations. 

1. 
2. 
3. 
4. 


2  +  3  +  5  —  ^^- 

Ans.  a:=:30. 

^-'=5+3 

a;  =15. 

X   .X       X        57 
4"^6~"10"~T' 

a:  =  45. 

3a:  +  4  — ^  =  46  — 2a:. 

a;  =  9. 

3-  +  5a;  +  3  =  28  +  -- 

6 
7* 

a:  =  4. 

|--  =  39-5.  +  |- 

5 

"8* 

a:«=9. 

|  +  4  =  -  +  12-- 

a:=13H. 

^x       7a:   ,    3a;   '    7a: 

a:  =  66|. 

1-1+10=1-1  +  11. 

a:  =12. 

4a:       3x       7a:       13a:       ^x 

.   '5 

5  "^  4        3  "~  10         2 

+'!• 

a:  =  3. 

7. 
8. 
9. 

10. 

The  equations  above  have  been  taken  at  random.  An  equa- 
tion, however,  may  always  be  considered  as  derived  from  the 
enunciation  of  some  question.  Thus  the  first  of  the  above  equa- 
tions may  be  considered  as  derived  from  the  following  enunci- 
ation, viz.,  to  find  a  number  such  that  one  half,  one  third,  and 
one  fifth  of  this  number  may  together  be  equal  to  31. 

15.  Though  no  general  and  exact  rule  can  be  given  for  put- 
ting a  problem  into  an  equation,  yet  the  following  precept  will 
be  found  very  useful  for  this  purpose,  viz. :  Indicate  by  the  aid 
of  algebraic  signs  upon  the  unknown  and  known  quantities  the 
same  reasonings  and  the  same  operations,  that  it  would  be  neces- 
sary to  perform  in  order  to  verify  the  answer,  if  it  were  known. 

Let  us  illustrate  this  precept  by  some  examples. 


EQUATIONS   OF   THE    FIRST   DEGREE.  15 

1.  A  gentleman  distributing  money  wanted  10  shillings  to  be 
able  to  give  5  shillings  to  each  person ;  he  therefore  gave  each  4 
shillings  only  and  found  that  he  had  5  shillings  left.  Required 
the  number  of  persons. 

In  order  to  verify  the  answer  if  it  were  known,  we  should 
multiply  it  first  by  5  and  from  the  product  subtract  10;  we 
should  next  multiply  it  by  4  and  add  5  to  the  product.  The 
results  thus  obtained  would  be  equal  to  each  other,  if  the  answer 
were  correct. 

Let  us  indicate  the  same  operations  by  the  aid  of  algebraic 
signs.  Putting  x  for  the  number  of  persons  sought  a-nd  multi- 
plying a:  by  o  we  have  5x,  subtracting  10  from  this  we  have 
5z  —  ]j3 ;  again  x  multiplied  by  4  gives  4a;,  adding  5  to  this  we 
have  4:X-\-5.  Then  as  these  two  results  should  be  equal  we 
have  for  the  equation  of  the  problem 

5x—10  =  Ax-\-5, 
which  being  resolved  gives  x  =  15. 

2.  A  person  expends  the  third  part  of  his  income  in  board  and 
lodging,  the  eighth  part  in  clothes  and  washing,  the  tenth  part  in 
incidental  expenses,  and  yet  saves  $318  yearly.  What  is  his 
yearly  income  ?  Ans.  $720. 

3.  A  and  B.  began  to  play ;  A  with  exactly  four-ninths  the 
sum  B  had.  After  A  had  won  $  10,  he  found  that  they  had  each 
the  same  sum.     What  had  A  at  first  ?  Ans.  $  16. 

4.  A  General  having  lost  a  battle  found  that  he  had  only 
3600  men  more  than  half  his  army  left,  fit  for  action ;  600  more 
than  one-eighth  of  his  men  being  wounded,  and  the  rest,  which 
were  one-fifth  of  the  whole  army,  either  slain,  taken  prisoners  ox 
missing.     Of  how  many  men  did  his  army  consist? 

Ans.  24,000. 

5.  A  sum  of  money  was  to  be  divided  among  six  poor  per- 
sons; the  second  received  lOd.  the  third  IM.  the  fourth  25d. 
the  fifth  28i.  and  the  sixth  23d.  less  than  the  first.  Now  the 
Bum  distributed  was  lOd.  more  than  the  treble  of  what  the  first 
received.    What  money  did  the  fijrst  receive  ?  Ans.  40rf. 


16  ELEMENTS    OF    ALGEBRA. 

6.  A  father  intends  by  his  will  that  his  three  sons  should 
share  his  property  in  the  following  manner.  The  eldest  is  to 
receive  100  pounds  less  than  half  the  whole  property,  the  second 
is  to  receive  80  pounds  less  than  a  third  of  the  whole  property, 
and  the  third  is  to  have  60  pounds  less  than  a  fourth  of  the 
property.  Required  the  amount  of  the  whole  property,  and  the 
share  of  each  son. 

7.  A  cistern  is  supplied  by  two  pipes,  the  first  will  fill  it  alone 
in  three  hours,  the  second  in  four  hours.  In  what  time  will  the 
cistern  be  filled  if  both  run  together  ? 

If  the  time  were  known,  we  should  verify  it  by  calculating 
what  part  of  the  cistern  would  be  filled  by  each  pipe  separately ; 
these  parts  added  toother  would  be  equal  to  the  whole  cistern. 
To  indicate  the  same  operations  by  the  aid  of  algebraic  signs, 
let  a;  =  the  time,  and  let  the  capacity  of  the  cistern  be  repre- 
sented by  1.     It  is  evident  that  if  one  of  the  pipes  will  fill  the 

cistern  in  three  hours,  in  one  hour  it  will  fill  -  of  it,  in  x  hours  it 

X 

will  fill  X  times  as  much,  that  is,  a  part  denoted  by  jr.     In  like 

o 

manner  in  the  time  x^  the  second  pipe  will  fill  a  part  denoted 

X 

by  -r ;  since  then  these  two  parts  should  be  equal  to  the  whole 
cistern,  we  have  for  the  equation  of  the  problem 

-4-- — 1 

from  which  we  obtain  x=.\\  hours. 

8.  A  cistern  is  furnished  with  three  cocks,  the  first  will  fill  it 
in  5  hours,  the  second  in  13  hours,  and  by  the  third  it  would  be 
emptied  in  9  hours.  In  what  time  will  the  cistern  be  filled  if  all 
three  run  together  ?  Ans.  6^^j  hours. 

9.  A  gentleman  having  a  piece  of  work  to  do  hired  three  men 
to  do  it ;  the  first  could  do  it  alone  in  7  days,  the  second  in  9, 
tjie  third  in  15  days.  How  long  would  it  take  the  three  together 
to  do  it  ?  Ans.  S^Vx  days. 


EQUATIONS    OF    THE    FIRST    DEGREE.  17 

10.  To  divide  the  number  247  into  three  parts,  which  may  be 
to  each  other  as  the  numbers  3,  5  and  11. 

Two  numbers  are  said  to  be  to  each  other  as  3  to  5,  or  in 
proportion  of  3  to  5,  when  the  first  is  three-fifths  of  the  second, 
or  which  is  the  same  thing,  when  the  second  is  five-thirds  of  the 
first. 

If  then  one  of  the  parts,  the  first  for  example,  were  known,  we 
should  verify  it  thus.  We  should  find  a  number,  which  would 
be  five-thirds  of  the  first  part;  this  would  be  the  second  part; 
we  should  find  also  a  number  which  would  be  eleven-thirds  of 
the  first  part;  this  would  be  the  third  part;  the  sum  of  these 
parts  would  then  be  equal  to  247. 

To  imitate  this  process  let  x  =  the  first  part,  the  second  will 

5x  11a: 

then  be  —  and  the  third  —^.     We  have  then  for  the  equation 
o  o 

of  the  question 

^  +  f +^  =  247, 
whence  x  =  39. 

11.  A  sum  of  money  is  to  be  divided  between  two  persons, 
A  and  B,  so  that  as  often  as  A  receives  9  pounds,  B  receives  4. 
Now  it  happens  that  A  receives  15  pounds  more  than  B.  What 
are  their  respective  shares?  Ans.  A  £27,  B  £12. 

12.  A  merchant  bought  a  piece  of  cloth  at  the  rate  of  7  crowns 
for  5  yards,  which  he  sold  again  at  the  rate  of  11  crowns  for  7 
yards,  and  gained  100  crowns  by  the  traffic.  How  many  yards 
were  there  in  the  piece  ?  Ans.  583-^  yds. 

13.  On  an  approaching  war  594  men  are  to  be  raised  from 
three  towns  A,  B,  C,  in  proportion  to  their  population.  Now 
the  population  of  A  is  to  that  of  B  as  3  to  5 ;  whilst  the  popula- 
tion of  B  is  to  that  of  C  as  8  to  7.  How  many  men  must  each 
town  furnish?  Ans.  A  144,  B  240,  C  210. 

14.  A  gentleman  employed  two  workmen  at  different  times, 
one  for  3  shillings,  and  the  other  for  5  shillings  a  day.     Now 

2 


18  ELEMENTS    OF   ALGEBRA. 

the  number  of  days  added  together  was  40 ;  and  they  each  re- 
ceived the  same  sum.     How  many  days  was  each  employed  ? 

If  the  number  of  days  one  of  the  workmen  was  employed, 
the  second  for  example,  were  known,  we  should  verify  it  thus, 
we  should  subtract  this  number  from  40,  this  would  give  the 
number  of  days  the  first  workman  was  employed;  multiplying 
next  the  number  of  days  the  first  workman  was  employed  by 
3,  and  that  of  the  second  by  5,  the  two  products  would  be 
equal. 

To  indicate  the  same  operations  let  x  =  the  number  of  days 
the  second  workman  was  employed,  then  40  —  x  will  be  the 
number  of  days  the  first  was  employed,  and  the  product  of 
40  —  X  multiplied  by  3  should  be  equal  io  x  y^  6. 

The  multiplication  of  40  —  a:  by  3  is  indicated  by  inclosing 
this  quantity  in  a  parenthesis  and  writing  the  3  outside,  thus, 
3  (40  —  x) ;  we  have,  therefore,  for  the  equation  of  the  ques- 
tion 

3  (40  — a;)  =  52:. 

With  respect  to  the  multiplication  required  in  this  equation,  it 
is  evident,  since  40  should  be  diminished  by  the  number  of  units 
in  x,  that  40  multiplied  by  3  would  be  too  great  for  the  product 
required,  by  the  number  of  units  in  x  multiplied  by  3 ;  to  obtain 
the  true  product  therefore  from  40  X  3,  we  must  subtract  a;  X  3 ; 
we  have  then 

120  — 3a;  =  5a:, 
from  which  we  obtain  a:  =  15. 

15.  Two  workmen  received  the  same  sum  for  their  labor; 
but  if  one  had  received  15s.  more,  and  the  other  9s.  less,  then 
one  would  have  had  just  three  times  as  much  as  the  other. 
What  did  they  receive  ?  Ans.  21s. 

16.  A  has  three  times  as  much  money  as  B ;  but  if  A  gains 
$  50  and  B  loses  S  93,  then  A  will  have  five  times  as  much  money 
as  B.     How  much  has  each?  Ans.  A  S772|,  B  $257|. 

17.  A  and  B  engaged  in  trade,  A  with  £240,  and  B  with 
£96.     A  lost  twice  as  much  as  B,  and  upon  settling  their  ac- 


EQUATIONS    OF  *THE    FIRST    DEGREE.  19 

counts  it  appeared  that  A  had  three  times  as  much  remaining 
as  B.    How  much  did  each  lose  ?  Ans.  A  £96,  B  £48. 

18.  Two  merchants  engage  in  trade,  each  with  the  same 
sum ;  A  gains  $  150,  B  loses  $  63j  when  it  appears  that  three 
times  A's  money  is  equal  to  five  times  B's.  What  had  each 
at  first?  Ans.  $382^. 

19.  A  laborer  was  hired  for  48  days ;  for  each  day  that  he 
wrought  he  was  to  receive  24  shillings,  but  for  each  day  that  he 
was  idle  he  was  to  forfeit  12  shillings.  At  the  end  of  the  time 
he  received  504  shillings.  How  many  days  did  he  work  and 
how  many  was  he  idle  ? 

To  verify  the  numbers  required  in  this  problem  we  should 
multiply  them,  if  known,  by  24  and  12  respectively ;  subtracting 
the  last  product  from  the  first,  the  remainder  would  be  504. 
To  indicate  these  operations  by  the  aid  of  algebraic  signs  let 
X  =  the  number  of  days  in  which  the  laborer  wrought,  then 
48  —  X  will  be  the  number  of  days,  in  which  he  was  idle;  24a: 
will  be  the  sum  due  for  the  number  of  days  in  which  he  wrought, 
and  576  —  12  a;  will  be  the  sum  which  he  forfeited. 

The  subtraction  of  576 — 12a;  from  24a;  is  indicated  by  in- 
closing this  quantity  in  a  parenthesis  and  writing  the  sign  — 
before  it,  thus,  24  a; — (576 — 12  a:);  we  have  then  for  the 
equation  of  the  question 

24a;— (576— 12a:)  =  504. 

To  perform  the  subtraction  required  in  this  equation,  it  is 
evident,  since  576  should  be  diminished  by  12  a;  before  subtrac- 
tion, if  we  take  576  from  24  a;  we  subtract  too  much  by  12  a;; 
12a;  must  therefore  be  added  to  this  result  in  order  to  have  the 
true  remainder ;  we  have  then 

24a;  —  (576  —  12  i)  =  24a;  —  576  +  12a;, 
the  equation  of  the  problem  therefore  becomes 

24a;  — 576+ 12a;  =  504, 
from  which  we  deduce  x  =  30. 

20.  A  father  being  questioned  as  to  the  age  of  his  son  replied, 


20 


ELEMENTS   OF   ALGEBRA. 


that  if  from  double  his  present  age,  the  triple  of  what  it  was  six 
years  ago  were  subtracted,  the  remainder  would  be  exactly  his 
present  age.     Kequired  his  age.  Ans.  9  years. 

21.  Divide  the  number  68' into  two  such  parts,  that  the  dif- 
ference between  84  and  the  greater  may  equal  three  times  the 
difference  between  40  and  the  less. 

Ans.     The  parts  will  be  26  and  42. 

22.  Two  men  commenced  trade ;  A  had  twice  as  much  money 
as  B;  A  gained  $50  and  B  lost  $90;  then  if  three  times 
B's  money  be  subtracted  from  A^s,  four  times  the  remainder 
will  be  exactly  equal  to  A's  money  at  first?  What  had  each 
at  first?  Ans.  A  $426f,  B  $213^. 

23.  A  person  at  play  won  as  much  as  he  began  with  and 
then  lost  18  shillings ;  after  this  he  lost  five-ninths  of  what  re- 
mained, and  then  counting  his  money,  he  found  he  had  14  shil- 
lings less  than  at  first.     What  had  he  at  first  ? 

Let  a:  =  the  number  of  shillings  he  began  with,  then  2x  will 
be  the  sum  he  had  after  winning  x,  and  2x — 18  the  sum  re- 
maining after  the  first  loss,  four-ninths  of  which  will  be  the 
sum  remaining  after  the  second  loss.     One-ninth  of  2x — 18 

is  expressed  thus, ,  ^ 

y 

four-ninths,  therefore,  will  be , 

y 

and  we  have  for  the  equation  of  the  question 

8a:  — 72       ,^ 

X 9— =14; 

from  which  we  obtain   9x  —  8  a;  -[-  72  =  126  ; 
whence  x  =  54c. 

24.  Divide  the  number  96  into  two  such  parts,  that  four-fifths 
of  the  greater,  diminished  by  three-fourths  of  the  less,  will  be 
equal  to  15.  Ans.    The  parts  are  56^-j-  aftgid  39f^. 

25.  It  is  required  to  divide  84  into  two  such  parts,  that  if 
one-half  of  the  less  be  subtracted  from  the  greater,  and  one- 


EQUATIONS   OF   THE   FIRST    DEGREE.  21 

eighth  of  the  greater  be  subtracted  from  the  less,  the  remainders 
shall  be  equal.  Ans.     The  parts  are  48  and  36. 

26.  A  and  B  began  to  trade  with  equal  sums  of  money.  In 
the  first  year  A  gained  40  pounds  and  B  lost  40 ;  but  in  the 
second  A  lost  one-third  of  what  he  then  had  and  B  gained  a  sum 
less  by  40  pounds  than  twice  the  sum  A  had  lost;  when  it 
appeared  that  B  had  twice  as  much  money  as  A.  What  money 
did  each  begin  with  ?  Ans.     £320. 

27.  What  two  numbers  are  as  3  to  5,  to  each  of  which  if  4  be 
added  the  sums  will  be  as  5  to  7  ? 

5x 

Let  X  =  the  less  number,  then  -k-=  the  greater ;  adding  4  to 

each,  the  first  will  be  a:  -|  4  and  the  second 

5x  ,    ^       5x  +  12 
-3-  +  4,or-:n_; 

but  by  the  question  seven-fifths  of  the  first  should  now  be  equal 
to  the  second,  we  have  therefore 

7{x  +  4)__5x+12 
5       '^       3      ' 

28.  Divide  the  number  49  into  two  such  parts,  that  the  greater 
increased  by  6  may  be  to  the  less  diminished  by  11  as  9  to  2. 

,.  Ans.     The  parts  are  30  and  19. 

29.  A  and  B  begin  trade,  A  with  triple  the  stock  of  B.  They 
gain  each  $50,  which  makes  their  stocks  in  the  proportion  of  7 
to  3.     Kequired  their  original  stocks. 

Ans.     A's  S300,  B's  100. 

30.  A,  B  and  C  make  a  joint  stock.  A  puts  in  $60  less  than 
B,  and  $68  more  than  C,  and  the  sum  of  the  shares  of  A  and  B 
is  to  the  sum  of  the  shares  of  B  and  C  as  5  to  4.  What  did 
each  put  in  ?  Ans.     A  $140,  B  $200,  and  C  $72. 

31.  A  man  being  at  play  lost  one  fourth  of  his  money  and  then 
won  3  shillings ;  after  which  he  lost  one  third  of  what  he  then 
had  and  won  2  shillings ;  lastly  he  lost  one  7th  of  what  he  then 
had ;  this  being  done  he  had  but  12  shillings  left.  What  had  he 
at  first?  Ans.     20s. 


22  ELEMENTS    OF    ALGEBRA. 

32.  There  are  three  pieces  of  cloth,  whose  lengths  are  in  the 
proportion  of  3,  5  and  7 ;  and  6  yards  being  cut  off  from  each, 
the  whole  quantity  is  diminished  in  the  proportion  of  20  to  17. 
Required  the  length  of  each  piece  at  first. 

Ans.     24,  40  and  56  yds. 

33.  Two  persons,  A  and  B  have  both  the  same  annual  income. 
A  lays  by  one-fifth  of  his ;  but  B  by  spending  £80  per  annum 
more  than  A,  at  the  end  of  4  years  finds  himself  £220  in  debt. 
What  did  each  receive  and  expend  annually  ? 

Ans.  Their  income  is  £125.     A  spends  £100,  B  £180. 

34.  A  man  bought  a  horse  and  chaise  for  S273.  Now  if 
three  fourths  the  price  of  the  horse  be  subtracted  from  the  price 
of  the  chaise,  the  remainder  will  be  equal  to  five-elevenths  the 
price  of  the  chaise  subtracted  from  four  times  the  price  of  the 
horse.     Required  the  price  of  each. 


SECTION  III.— Algebraic  Operations. 

16.  A  quantity  expressed  by  algebraic  signs  is  called  an  alge- 
braic of  literal  quantity.  Thus,  a-\-^lT-\-^x^al^xyz,  are 
algebraic  or  literal  quantities. 

From  what  has  been  done,  it  is  easy  to  see  that  we  shall 
have  frequent  occasion  to  perform  upon  algebraic  quantities 
operations  analogous  to  the  fundamental  operations  of  arith- 
metic, viz.  addition,  subtraction,  multiplication  and  division. 
The  operations  upon  algebraic  quantities,  differ  however  from 
the  corresponding  ones  in  arithmetic  in  this  respect,  that  the 
results  at  which  we  arrive  in  the  case  of  algebraic  quantities  are 
for  the  most  part  only  indications  of  operations  to  be  performed. 
All  that  we  do  is  to  transform  the  operations  originally  indicated 
into  others,  which  are  more  simple,  or  which  become  necessary 
in  order  that  the  conditions  of  the  question  may  be  fulfilled. 
Thus,  in  the  equation  a;  4-2 a; -f- 6a:  =  405,  given  by  the  con- 


ADDITION    OF    ALGEBRAIC    QUANTITIES.  23 

ditions  of  question  first  art.  6,  we  simplify  the  operations 
originally  indicated  by  reducing  the  expressions  x-\-2x'^6x 
to  one  term,  9x,  by  an  operation  analogous  to  addition  in  arith- 
metic, though  not  strictly  the  same.  So  likewise  in  qur,stion 
nineteenth,  art.  15,  though  we  cannot,  strictly  speaking,  subtract 
.576  —  12 z  from  24a:,  yet,  by  an  operation  analogous  to  subtrac- 
tion in  arithmetic,  we  indicate  upon  these  quantities  operations, 
which  produce  the  same  effect,  as  the  subtraction  which  the 
conditions  of  the  question  require. 

17.  Algebraic  quantities  consist'^ g  only  of  one  term  are  called 
Tnonomiah,  as  3  a,  —  4^,  &c.  Those  which  consist  of  two 
terms  are  called  hinomials^  as  a-\-h^  c  —  d.  Those  which  con- 
sist of  three  terms  are  called  trinomials,  &c.  In  general, 
quantities  consisting  of  more  than  one  term  are  called  poly- 
nomials. Quantities  consisting  only  of  one  term  are  also  called 
simple  quantities,  and  those  consisting  of  more  than  one  term 
are  called  compound  quantities. 

Quantities  in  algebra,  which  are  composed  of  the  same  letters, 
and  in  which  the  same  letters  are  repeated  the  same  number  of 
times,  are  called  similar  quantities,  thus,  3 ah,  1  ab  are  similar 
quantities,  so  also  aab,  5aab. 

ADDITION    OF   ALGEBRAIC    QUANTITIES. 

18.  1.  Let  it  be  required  to  add  the  monomials  a,  b,  c,  and 
d  ;  the  result,  it  is  evident,  will  be  a -\- b  -\-  c -\-  d. 

2.  Let  the  quantities  to  be  added  be  ab,  c,  ab,  d  Here  we 
have  as  before  ab  -\-  c  -{-  ab  -{-  d ;  but  the  quantities  ab,  ab  in 
this  result  are  similar,  they  may  therefore  be  united  in  one  term, 
thus,  2ab;  whence  the  sum  required  will  he  2ab  -^  c -\-  d.  To 
add  monomials  therefore,  Write  them  one  after  the  other  with 
the  sign  -\-  between  them,  observing  to  simplify  the  result  by 
uniting  in  one,  those  lohich  are  similar. 

3.  Let  it  next  be  required  to  add  the  polynomials  a-\-b  and 
c-\-  d-\-e.     The  sum  total  of  any  number  of  quantities  what- 


24  ELEMENTS    OF    ALGEBRA. 

ever  should  be  equal,  it  is  .evident,  to  the  sum  of  all  the  parts  of 
which  these  quantities  are  separately  composed ;  we  have  there- 
fore for  the  sum  required  a-\-b-\-c-\-d-\-e. 

Let  the  quantities  proposed  he  a-\-b  and  c  —  d.  If  we 
begin  by  adding  c,  the  result  a-\-b-\-c  will,  it  is  evident,  be 
too  great  by  the  quantity  d,  since  it  is  not  c,  which  we  are  to 
add,  but  c  diminished  by  d;  to  obtain  the  true  result,  therefore, 
from  a-\-b  -{-  c  we  must  subtract  d ;  whence  c  —  d  added  to 
a-\-  b  gives 

a-\-b  -\-  c  —  d. 

To  add  polynomials  therefore,  Write  in  order  one  after  the 
other  the  quantities  to  be  added  with  their  proper  signs,  it  being 
observed  that  the  terms,  which  have  no  signs  before  them,  are 
considered  as  having  the  sign  -{-. 

19.  Let  it  riext  be  required  to  add  the  following  quantities. 
^a-\-lib  —  2c 
2a  —  5c 
83 +  c. 

By  the  rule  just  given  the  sum  required  will  be 
Qa-{-7b  —  2c-\-2a  —  5c  +  8b-{-c. 

In  this  result  the  similar  terms  9  a,  2  a  may  be  united  in  one, 
II a;  also  the  terms  73  and  83  give  153. 

The  similar  quantities  — 2  c,  —  5  c  being  both  subtractive, 
the  effect  will  be  the  same,  if  we  unite  them  in  one  sum  7c 
and  subtract  this  sum;  and  as  there  would  still  remain  the 
quantity  c  to  be  added,  instead  of  first  subtracting  7  c  and  then 
adding  c  to  the  resuk,  the  effect  will  be  the  same  if  we  subtract 
only  6  c. 

The  sum  of  the  expressions  proposed  will  then  be  reduced  to 
.la-fl53  — 6c. 

In  order  to  verify  this  result,  let  us  put  numbers  for  the  letters 
a,  3,  c,  in  the  proposed  •  for  example,  the  numbers,  10,  4,  - 
respectively,  and  we  have 


ADDITION   OF   ALGEBRAIC    QUANTITIES.  X6 

9a-{-7b'-2c=112 
2a  —  5c=5 
8b  +   c=   25 

9a_|-7i  — 2c  +  2a  — 5c +  8^  + c==  152 
Making  the   same  substitution  in  the   expression   lid -{-15b 
—  t5c,  we  obtain  the  same  result. 

The  operation,  by  which  all  similar  terms  are  reduced  to  one, 
whatever  sign  they  may  have,  is  called  reduction.  To  perform 
this  operation,  Take  ike  sum  of  similar  quantities,  which  have 
the  sign  -j-  a7id  that  of  those  ivhich  have  the  sign  —  ;  subtract 
the  less  of  the  two  sums  from  the  greater  and  give  to  the  remain' 
der  the  sign  of  the  greater. 

We  have  then  the  following  general  rule  for  the  addition  of 
algebraic  quantities,  viz.  Write  the  quantities  in  order  one  after 
the  other  with  their  proper  signs,  observing  to  simplify  the  result 
by  reducing  to  one,  terms  which  are  similar. 


EXAMPLES. 


L  To  add  the  quantities 
5x-{-3y  —  4z 
Qz-\-2x^5y-\-2t 
35  — 42/  — 2z+a: 
7a;  — 3z  +  4y  — 65 


Answer  15a;  — 2?/- 3z4-2?  — 35. 

To  verify  this  answer  let  the  numbers  12,  5,  4, 3, 13,  be  put  for 
the  letters  x,  y,  z,  t,  s,  respectively. 
2.  To  add  the-quantities 

7^  +  3^_14;7  +  17r 
3a4-9?i— ll7w  +  2r 

5y 4772 -f-8^ 

lln — 2b  —  m  —  r-|-5 

Answer    31%- 97/1- 9p+ 18r +  3a  — 23  +  5. 
c 


26  ELEMENTS    OF    ALGEBRA. 

3.  To  add  the  quantities 

llbc-{-4.ad  —  Sac-\-5cd 
8ac-\-7hc  —  2ad-}- 4:77171 
2c d  —  3ab-\-  5ac-\-am 
'9  am  —  2b  c  —  2ad-\-5cd 

Answer  Wbc-{- 5ac~{-  l2cd-\-4mn — '3ab-\-  IQam, 


SUBTRACTION   OF   ALGEBRAIC    QUANTITIES. 

20.  1.  To  subtract  a  from  b.  Here  the  quantities  being  din 
similar,  the  subtraction  can  only  be  expressed  by  the  sign  — 
thus,  b  —  a. 

2.  To  subtract  5  a  from  7«.  The  quantities  in  this  case  being 
similar,  the  subtraction  may  be  performed  by  means  of  the  coef- 
ficients, and  the  result,  it  is  evident,  will  be  2  a. 

3.  To  subtract  2  3  -}-  3  c  from  d.  To  subtract  one  quantity 
from  another,  we  must,  it  is  evident,  take  from  this  other  the 
sum  of  all  the  parts,  of  which  the  quantity  to  be  subtracted  is 
composed.     The  result  required  will  therefore  be 

d  —  2b'-3c. 

4.  To  subtract  a  —  b  from  c.  If  we  begin  by  subtracting  a 
from  c,  it  is  evident,  that  we  shall  take  away  too  much  by  the 
quantity  b,  by  which  a  should  be  diminished  before  its  subtrac- 
tion ;  b  should  therefore  be  added  to  c  —  a  to  give  the  true  result ; 
whence  a  —  b  subtracted  from  c  gives 

c  —  a-\-b. 

5.  To  subtract  5c  +  2d  —  4tb  from  70  — 2d  — 5b.  The 
result,  it  is  easy  to  see,  will  be 

70  — 2d  — 5b  — 5c  — 3d +  U, 
which  becomes  by  reduction 

2c  —  5d  —  b. 
From  what  has  been  done  the  following  rule  for  the  subtrac- 
tion of  algebraic  quantities  will  be  readily  inferred,  viz.    Change 


MULTIPLICATION   OF   ALGEBRAIC    QUANTITIES.  S7 

the  signs  -\-  into  — ,  and  the  sig'os  —  into  -[-  in  the  quantities 
to  be  subtracted,  or  suppose  them  to  be  changed^  and  then  proceed 
as  in  addition. 

EXAMPLES. 

To  subtract  from  17a  +  2w— -93— 4c  +  23f^ 
the  quantity  51 «  —  273  +  llc  —  ^d 

Answer  27/i  — 34a  +  18Z»  — 15c  +  27  d. 

2.  To  subtract  from  5 ac  —  Sab-\-^bc  —  4am 
the  quantity  8am  —  2ab-\-llac  —  7cd 

Answer  9bc  —  6ac — 6ab — 12am-\-7cd 

3.  To  subtract  from 

15abc—13xy-\-2lcd  —  4.lx  —  25 
the  quantity  75xy  —  4:abc-\-16x  —  5Scd  —  Slmc 

Answer  19abc  —  88xy  —  57x-{-74:cd-{-Slmc  —  25 

MULTIPLICATION   OF   ALGEBRAIC    QUANTITIES. 

21.  1.  The  product  of  a  quantity  a  by  another  quantity  b  is 
expressed,  as  we  have  already  seen,  thus,  ay,b,  or  in  a  more 
simple  manner,  thus,  ab.  In  like  manner  the  product  of  abhy 
cd  is  expressed  thus,  abXcd,  or  thus,  abed. 

2.  The  letters  a  and  b  are  called  factors  of  the  product  ab. 
So  also  a,  b,  c  and  d  are  the  factors  of  the  product  abed.  The 
value  of  a  product,  it  is  easy  to  see,  does  not  depend  at  all  upon 
the  order,  in  which  its  factors  are  arranged ;  thus  the  value  of 
the  product  arising  from  the  multiplication  of  a  by  ^  will  evidently 
be  the  same,  whether  we  write  ba  or  ab. 

3.  Let  it  be  proposed  to  multiply  Sab  hj  5cd;  by  no.  1  we 
have  Sab  5cd,  or  by  no.  2,  3  X  5abcd;  but  the  factors  3  and  5 
in  this  result  may,  it  is  evident,  be  reduced  to  one  by  multiplying 
them  together ;  performing  this  operation,  the  product  required 
will  be  15 abed.  In  like  manner  the  product  of  the  quantities 
7ab,9cdf  13 e/ will  be 

Sldabcdef. 


28  ELEMENTS    OF    ALGEBRA. 

4.  Let  it  be  required  to  multiply  aahy  a.  According  to  no. 
1  we  have  for  the  result  aaa;  but  this  expression  for  the  product 
required  may,  it  is  easy  to  see,  be  abridged  by  writing  the  letter 
a  but  once  only,  and  indicating  by  a  figure  the  number  of  times 
this  letter  enters  into  it  as  a  factor.  The  figure  which  indicates 
the  number  of  times  a  given  letter  enters  as  a  factor  in  a  product 
is  called  the  exponent  of  that  letter.  And  in  order  to  distinguish 
the  exponent  of  a  letter  from  a  coefficient,  we  place  the  exponent 
at  the  right  hand  of  the  letter  and  a  little  above  it,  the  coefficient 
being  always  placed  before  the  letter,  to  which  it  belongs,  and  on 
the  same  line  with  it. 

According  to  this  method  the  product  ca  is  expressed  by  a^, 
aaa  by  a^,  aaaa  by  a*,  &c. 

A  letter,  which  is  multiplied  once  by  itself,  or  which  has  two 
for  an  exponent,  is  said  to  be  raised  to  the  second  power.  A 
letter  which  is  multiplied  twice  successively  by  itself,  or  which 
has  3  for  an  exponent  is  said  to  be  raised  to  the  third  power. 
In  general,  the  power  of  a  letter  is  designated  according  to  the 
figure,  which  it  has  for  an  exponent,  thus  a  with  7  for  an  expo- 
nent is  called  the  seventh  power  of  a. 

A  letter  which  has  no  exponent  is  considered  as  having  unity 
for  its  exponent,  thus  a  is  the  same  as  a^ 

From  what  has  been  said,  it  will  be  perceived,  that  in  order  to 
raise  a  letter  to  a  given  power,  it  is  necessary  to  multiply  it  suc- 
cessively by  itself  as  many  times  less  one  as  there  are  units  in  the 
exponent  of  this  power. 

5.  Let  it  next  be  required  to  multiply  a^  by  a^.  According 
to  no.  1  the  product  will  be  expressed  by  a^  a^.  In  this  pro- 
duct the  letter  a,  it  will  be  observed,  occurs  three  times  as  a 
factor,  and  also  five  times  as  a  factor,  whence  on  the  whole  it 
is  found  eight  times  as .  a  factor.  The  product  a^  a^  may  there- 
fore according  to  no.  4  be  expressed  more  concisely,  thus,  a^. 
In  like  manner  the  product  of  a'  by  a'  will  be  <z'^  Whence,  in 
general,  The  product  of  two  powers  of  the  same  letter  will  have 


MULTIPLICATION    OF   ALGEBRAIC    QUANTITIES.  29 

for  an  exponent  the  sum  of  the  exponents  of  the  multiplier  and 
multiplicand. 

6.  Let  it  be  proposed  next  to  multiply  a^  h"^  c  by  a*  b^  (?  d. 
According  to  no.  1  the  product  will  be  a^  h^  c  «*  b^  c^  d,  or  by 
no.  2,  a^  a*  b^  b^  c  c^  d ;  but  this  expression  may  be  reduced  by 
the  rule  just  given  to  a^  b^  c^  d ;  whence 

a'  b''  c  X  aHU^  d  =  an^  c^  d. 

From  what  has  been  done  we  have  the  following  rule  for 
the  multiplication  of  simple  quantities,  viz.  1°.  Multiply  the 
coefficients  together;  2°.  write  in  order  in  the  product  thus 
obtained  the  letters  ivhich  are  found  at  once  in  both  the  multiplier 
and  multiplicand,  observing  to  give  to  each  letter  the  sum  of  the 
exponents,  with  which  this  letter  is  affected  in  the  two  factors ; 
3°.  if  a  letter  is  found  in  one  of  the  factors  only,  write  it  in  the 
product  with  the  exponent  lohich  it  has  in  this  factor. 

EXAMPLES. 

To  multiply  1.  SaHc'hy  lahc^.         Ans.  Sea^iVdf. 

2.  21tt3^2crfby8a3c^       Ans.  168a* J^cV. 

3.  MabUh^^df  Ans.  W^abUdf 
22.  Let  us  pass  to  the  multiplication  of  polynomials. 

To  indicate  that  a  polynomial  a-\-b,  for  example,  is  multi- 
plied by  another  c-\-d,  we  draw  a  vinculum  over  each  and 
connect  them  by  the  sign  of  multiplication,  thus, 


or,  which  is  the  better  method,  we  inclose  each  of  the  quantities 
in  a  parenthesis  and  write  them  in  order  one  after  the  other, 
either  with  or  without  a  sign  of  multiplication,  thus, 
(«  +  *)X(c  +  e^),or(a  +  3)(c  +  ^). 

1.  To  multiply  a-\-b\y^  c.  To  form  the  product  required,  it 
is  evident,  that  we  must  take  c  times  each  of  the  parts  a  and  h 
of  which  the  quantity  a-\-b\%  composed. 

The  product  of     a-\-h 
multiplied  by  c 

is  therefore  ac-\-hc. 


30  ELEMENTS    OF    ALGEBRA. 

In  like  manner  2a-\-Pc-\-d 

multiplied  by  k 


gives  2ah-\-b^ch-\-dk. 

2.  To  multiply  a  —  bhy  c.  Since  a  —  3  is  smaller  than  a  by 
the  quantity  b,  ac  the  product  of  a  by  c,  it  is  evident,  will  be  too 
large  for  the  product  required  by  b  times  c  or  be;  whence  to 
obtain  the  true  result,  from  ac  we  must  subtract  be. 

The  product  pf  a  —  b 

multiplied  by  c 

is  therefore  ac  —  be 

In  like  manner  a^-^-  c^  —  dh  —  ef 

multiplied  hj  ah 


gives  a^k-\-ahc^  —  adk^  —  ahef. 

From  what  has  been  done,  it  is  evident,  that.  If  two  terms 
each  affected  with  the  sign  -f-  be  multiplied  together,  the  product 
must  have  the  sign  -\-  ;  but  if  one  of  the  terms  be  affected  icith 
the  sign  -\-  and  the  other  with  the  sign  — ,  the  product  must  have 
the  sign  — . 

3.  Let  it  be  proposed  next  to  multiply  a  —  3  by  c  —  d.  In 
this  case,  it  is  evident,  that,  if  we  take  c  times  a  —  5  the  result 
will  be  too  great  by  d  times  a  —  b  ;  whence,  to  obtain  the  true 
product,  from  c  times  a  —  b,  or,  ac  —  be,  we  must  subtract  d 
times  a  —  b  ox  ad  —  bd, 

The  product  of  a  —  b 

multiplied  by  c  —  d 

is  therefore  ac  —  bc' —  ad-\-bd. 

From  this  example  it  appears  that.  If  two  terms  be  affected 
each  with  the  sign  — ,  the  product  of  these  terms  should  be  affected 
with  the  sign  -|-. 

If  in  the  expression  'of  a  product  there  occur  similar  terms, 
the  expression  may  be  abridged  by  uniting  these  terms  into 
one. 


MTTLTIPLICATION   OF   ALGEBRAIC    QUANTITIES.  31 


Thus  2a^.2  +  a'  — ( 

multiplied  by   c^  —  ah^  •\'  & 


gives  aH^-\-a'  —  2a!'b*-{-3ab'c'  —  c'. 

To  verify  this  result  let  «  =  5,  b  =  2,  c  =  3. 

From  what  has  been  done  we  have  the  following  rule  for  the 
multiplication  of  polynomials,  viz.  1°.  Multiply  each  term  of  the 
multiplicand  by  each  term  of  the  multiplier,  observing  with 
respect  to  the  signs,  that  if  tivo  terms  multiplied  together  have 
each  the  same  sign,  the  product  must  have  the  sign  -\-,  but  if  they 
have  different  signs,  the  product  must  have  the  sign  — ;  2°.  Add 
together  the  partial  products  thus  obtai?ied,  taking  care  to 
unite  in  one,  terms  which  are  similar. 

23.  A  polynomial  is  said  to  be  arranged  with  reference  to 
some  letter,  when  its  terms  are  written  in  order  according  to  the 
powers  of  this  letter.     The  polynomial 

a^b^  +  a'b  —  ab'+a'b^ 
for  example,  arranged  in  descending  powers  of  the  letter  a  stands 
thus,  a*  b"^ -\- a^  b -{- a^  P  —  ab^;   arranged  in  ascending  powers 
of  the  letter  b  it  stands  thus,  a^ b -{- a*  b^ -\-  a^b^  —  abK 

The  letter  with  reference  to  which  the  arrangement  is  made  is 
called  the  principal  letter. 

To  facilitate  the  multiplication  of  polynomials,  it  is  usual, 
1°.  to  arrange  the  quantities  to  be  multiplied  according  to  the 
powers  of  the  same  letter;  2°.  to  dispose  of  the  partial  pro- 
ducts in  such  a  manner  that  those  terms,  which  are  similar, 
shall  fall  under  each  other.  Let  it  be  proposed,  for  example,  to 
multiply 

^3 ^Pa-{-a'  +  banYU^  —  3ba-{-  2a\ 

The  multiplier  and  multiplicand  being  both  arranged  with 
reference  to  the  letter  a,  the  work  will  be  as  follows : 


32  ELEMENTS    OF   ALGEBRA. 


Sa'  +  Sba'  +  Sb^a'^-Sb'a' 

—  Sba'  —  Sb^a^  —  Sb'a'  —  Sb'a 

U^a'  +  Wa'  +  Wa  +  W 
Sa'-\-4.b^a'  +  4.b'a'-{-b'a  +  4:b'. 

24.  The  following  examples  will  serve  as  an  exercise  in  the 
multiplication  of  polynomials. 

To  multiply 

1.  5a^  —  ^a'b  +  5ab''  —  2b^ 
by  4,a^^5ab  +  2b^  ' 

Answer  20a'  —  4:la'b  +  50a'b^  —  A5a^b^'}-25ab'  —  6b'. 

2.  a^  +  Sa'b  +  Sab^  +  b^ 

by  fl3_3^2^_j_0^^2_^3 

Answer  a^  —  2  a' b"" -\- 3  aH' —  b\ 

3.  x^  +  x^y  +  x'f  +  xy^  +  ?/* 
by          a;  —  7/ 

Answer  a:^  —  'if. 

25.  A  term  which  contains  one  literal  factor  only,  is  said  to 
be  of  the  first  degree  ;  a  term  which  contains  two  literal  factors 
only,  is  said  to  be  of  the  second  degree,  &c.  In  general,  the 
degree  of  a  term  is  marked  by  the  number,  which  expresses  the 
sum  of  the  exponents  of  the  letters,  which  enter  into  this  term. 
The  coefficient  is  npt  reckoned  in  estimating  the  degree  of  the 
term.  Thus  c^b^c  is  a  term  of  the  6th  degree,  and  7«^^  is  a 
term  of  the  fourth  degree. 

A  polynomial  is  said  to  be  homogeneous  when  all  its  terms  are 
of  the  same  degree.  Thus,  3c^  —  ^ab,^d? -\-abc  —  W  are 
homogeneous  polynomials. 

26.  From  the  rules  for  multiplication,  which  have  been  laid 
down,  it  follows, 

1°.  If  the  polynomials  proposed  for  multiplication  are  each 
homogeneous,  the  product  of  these  polynomials  will  also  he  ho* 


MULTIPLICATION    OF   ALGEBRAIC    QUANTITIES.  33 

mogeneouSi  and  the  degree  of  each  term  of  the  product  will  be 
equal  to  the  sum  of  the  degrees  of  any  two  terms  whatever  of  the 
multiplier  and  multiplicand.  Thus  in  the  first  example,  art.  24, 
all  the  terms  of  the  multiplicand  being  of  the  third  degree  and 
those  of  the  multiplier  of  the  second  degree,  all  the  terms  of  the 
product  are  of  the  fifth  degree.  When  therefore  the  factors  of  a 
product  are  homogeneous,  we  may  readily  detect  by  means  of 
this  remark  any  error  in  regard  to  the  exponents,  which  may 
have  occurred  in  the  course  of  the  work. 

2°.  In  the  multiplication  of  polynominals,  if  there  be  no  re- 
duction of  similar  terms,  the  number  of  terms  in  the  'product  will 
he  equal  to  the  number  of  terms  in  the  multiplicand  multiplied  by 
the  number  of  terms  in  the  multiplier.  Thus  if  there  be  5  terms 
in  the  multiplicand  and  4  in  the  multiplier,  there  will  be  20  in 
the  product. 

3".  But  if  there  be  a  reduction  of  similar  terms,  then  the 
number  of  terms  in  the  product  may  be  much  less.  It  should 
be  observed,  however,  that  among  the  different  terms  of  the 
product  there  will  be  two  at  least,  which  will  not  admit  of 
reduction  with  any  other,  viz.  1".  The  term  arising  from  the 
multiplication  of  the  term  in  the  multiplicand  affected  loith  the 
highest  exponent  of  one  of  the  letters,  by  the  term  in  the  multiplier 
affected  with  the  highest  exponent  of  the  same  letter.  2°.  The  term 
arising  from  the  multiplication  of  the  tioo  terms  affected  luith  the 
lowest  exponent  of  the  same  letter. 

The  manner  in  which  an  algebraic  product  is  formed  by  means 
of  its  factors  is  called  the  law  of  this  product.  This  law,  it  will 
readily  be  perceived,  remains  always  the  same,  whatever  may  be 
the  values  attributed  to  the  letters  which  enter  into  the  factors. 

27.  A  product  being  given,  we  may  sometimes  by  mere  in- 
spection decompose  it  into  its  factors,  an  operation  which  is 
frequently  useful. 

Let  there  be  the  product  a^h  —  a^b^.  In  the  formation  of 
this  product  each  term,  it  is  evident,  has  been  multiplied  by  a* 


34  ELEMENTS    OF   ALGEBRA. 

and  also  by  h,  its  factors  therefore  are  c^,  h  and  a  —  3,  and  it 
may  be  put  under  the  form  c^b  {a  —  b). 

In  like  manner  the  product  ac-\-ad-\-bC'^bd  may  be 
put  under  the  form  a{C'\- d) -\-b  {c-\- d)^  or  which  is  the 
same  thing  {a  -\-b)  {c-\-d). 


DIVISION   OF   ALGEBRAIC    QUANTITIES. 

28.  1.  The  object  of  division  in  algebra  is  the  same  as  that 
of  division  in  arithmetic,  viz.  to  find  one  of  the  factors  of  a  given 
product^  token  the  other  is  known. 

According  to  this  definition  the  divisor  multiplied  by  the  quo- 
tient must  produce  anew  the  dividend;  the  dividend,  therefore, 
must  contain  all  the  factors  both  of  the  divisor  and  quotient; 
whence  the  quotient  is  obtained  by  striking  out  of  the  dividend 
the  factors  of  the  divisor. 

Thus  to  divide  abed  by  ac,  we  strike  out  of  the  dividend 
the  factors  a  and  c  of  the  divisor  and  obtain  bd  for  the  quo- 
tient. 

2.  Let  it  be  required  to  divide  a^b  by  a^b.  Decomposing  a^ 
into  the  two  factors  a^  and  a^,  the  dividend  may  be  put  under  the 
form  a^a^b;  whence  striking  out  of  the  dividend  the  factors 
a^  and  b  of  the  divisor,  the  quotient  will  be  a^. 

From  this  example  it  appears  that  in  order  to  find  the  quotient 
of  two  powers  of  the  same  letter ;  From  the  exponent  of  the  divi' 
dend  lae  subtract  that  of  tke  divisor,  ike  remainder  loill  be  the 
exponent  of  the  quotient. 

3.  If  it  be  required  to  divide  72  a  J' c  by  9^.^  we  find  that  72, 
the  coefficient  of  the  dividend,  may  be  decomposed  into  the  two 
factors  9  and  8 ;  V  may  also  be  decomposed  into  the  two  factors 
b^  and  b"^',  the  dividend  therefore  may  be  put  under  the  form 
9  X  8ab^b^c;  whence,  suppressing  9  and  P,  the  factors  of  the 
divisor,  we  have  8ab^c  for  the  quotient. 

From  what  has  been  said  we  have  the  following  rule  for  the 
division  of  simple  quantities,  viz.    P.  Divide  the  coefficient  of 


DIVISION    OF   ALGEBRAIC    QUANTITIES.  35 

the  dividend  by  the  coefficient  of  the  divisor  ;  2°.  suppress  in  the 
dividend  the  letters,  ivhich  are  common  to  it  and  the  divisor,  when 
they  have  the  same  exponent,  and  when  the  exponent  is  not  the 
same,  subtract  the  exponent  of  the  divisor  from  that  of  the  divi- 
dend and  the  remainder  will  be  tlve  exponent  to  be  affixed  to  the 
letter  in  the  quotient ;  2°.  write  in  the  quotient  the  letters  of  tJte 
dividend,  which  are  not  in  the  divisor. 

EXAMPLES. 

1.  To  divide  48an'c''d  by  I2ab^c.  Ans.  Aa'bcd. 

2.  To  divide  150a'b'c^  by  SOa'b'd!'.  Ans.  5aH'cd. 

29.  From  the  preceding  rule,  it  is  evident,  in  order  that  the 
division  may  be  possible,  1°.  that  the  coefficient  of  the  divisor 
should  exactly  divide  the  coefficient  of  the  dividend;  2".  the 
exponent  of  a  letter  in  the  divisor  should  not  exceed  the  expo- 
nent of  the  same  letter  in  the  dividend ;  3°.  that  there  should  be 
no  letter  in  the  divisor,  which  is  not  found  in  the  dividend. 

When  these  conditions  do  not  exist,  the  division  can  only 
be  indicated  by  the  usual  sign.  If  it  be  required,  for  example, 
to  divide  12  a^b  by  9cd,  the  division,  it  is  easy  to  see,  can- 
not be  performed;  we  therefore  express  the  quotient  by  WTit- 
ing  the  divisor  under  the   dividend  in  the  form  of  a  fraction, 

l2aH 
thus,  -Q— 7-. 
9cd 

30.  The  expression  — — -  is  called  an  algebraic  fraction. 

9cd 

Fractions  of  this  species  may  be  simplified,  in  the  same  manner 

as  those  of  arithmetic,  by  striking  out  the  factors,  which  are 

common  to  both  terms,  or  which  is  the  same  thing,  by  dividing 

both  terms  by  the  factors,  which  are  common  to  them. 

Let   it   be    required,   for    example,   to   divide    48a^b^cd^  by 

SQa^Pc^de;  from  what  has  been  said,  the  most  simple  expres- 

•n  1-     4a^ 
sion  for  the  quotient  will  be  ^ — . 

In  like  manner  a^b  divided  by  5a^b  gives  ^  for  the  quo- 
tient. 


38  t  ELEMENTS    OF   ALGEBRA. 

31.  It  scwne times  ha,ppens,  that  the  exponent  of  a  letter  ia 
the  same  both  in  the  divisor  and  dividend.  The  rule  for  ob- 
taining the  exponents  of  the  letters  of  the  quotient,  art.  28,  being 
applied  to  a  case  of  this  kind,  will  give  zero  for  the  exponent 

of  the  letter  in  the  quotient.  Thus,  -^  according  to  this  rule 
gives  aP  for  a  quotient ;  but  -g,  it  is  evident,  is  equal  to  unity ; 

the  expression  a°  may  therefore  be  considered  as  equivalent 
to  unity.  In  general,  a  letter  with  zero  for  an  exponent  is  to  he 
regarded  as  a  symbol  equivalent  to  unity. 

This  symbol,  it  is  evident,  will  produce  no  effect  upon  the 
value  of  the  expression,  in  which  it  appears  as  a  factor,  since  it 
signifies  nothing  but  unity.  Its  only  use  is  to  preserve  in  the 
work  the  trace  of  a  letter,  which  formed  a  part  of  the  question 
proposed,  but  which  would  otherwise  disappear  by  the  efTect  of 
division.  Thus,  if  it  be  required  to  divide  24  a^i*^  by  Sd^h^,  the 
quotient  from  what  has  been  said  may  be  put  under  the  form 
3<zi°.  The  symbol  b°  indicates  that  the  letter  b  enters  0  times 
as  a  factor  in  this  result,  or  in  other  words  that  it  does  not  enter 
into  it  as  a  factor,  but  at  the  same  time  it  serves  to  show  that 
this  letter  belonged  as  a  factor  to  the  quantities,  from  which  the 
result  3  a  is  obtained  by  division. 

32.  We  pass  next  to  the  division  of  polynominals.  Since  the 
divisor  multiplied  by  the  quotient  should  produce  anew  the  divi- 
dend, it  is  evident,  that  the  dividend  must  contain  all  the  partial 
products  arising  from  the  multiplication  of  each  term  of  the 
divisor  by  each  term  of  the  quotient.  This  being  the  case,  it 
is  easy  to  see,  that  if  we  can  find  any  one  of  these  partial 
products  in  the  dividend,  and  the  particular  term  of  the  divisor 
upon  which  it  depends  is  known,  by  dividing  this  term  in  the 
dividend  by  the  known  term  of  the  divisor,  we  shall  obtain  a 
term  of  the  quotient  sought. 

Let  it  be  required  to  divide 

50fl3*''  — 41a*i-|-20a5+10ai*  — 33a»i' 
by        5ah^  —  ^a'b'\-5aK 


DIVISION   OF   ALGEBRAIC    QUANTITIES.  37 

It  is  evident  from  what  has  been  said,  art.  26,  that  the  term 
fl^,  being  affected  with  the  highest  exponent  of  the  letter  a  in 
the  dividend,  must  have  been  formed  without  any  reduction 
from  the  multiplication  of  5  a?,  the  term  affected  with  the  high- 
est exponent  of  the  letter  a  in  the  divisor,  by  the  term  affected 
with  the  highest  exponent  of  the  same  letter  in  the  quotient; 
that  is,  the  term  20  a^  of  the  dividend  is  the  product  of  5  a?  of  the 
divisor  by  a  term  of  the  quotient;  whence,  dividing  20 a^  by  6c^ 
we  obtain  ^d?  one  of  the  terms  of  the  quotient  sought.  Multi- 
plying the  divisor  by  4a^,  we  produce  anew  all  the  terms  of  the 
dividend,  which  depend  upon  4fl^  viz.  20  a?  b^ —  \Q  a^  b -\- 20  c^  \ 
subtracting  these  from  the  dividend,  the  remainder 

^0a?b''-^25a'b-{-\0ab'-^^^aH^ 
must  contain  all  the  partial  products  arising  from  the  multipli- 
cation of  each  one  of  the  remaining  terms  of  the  quotient  by 
each  term  of  the  divisor. 

Regarding  this  remainder  as  a  new  dividend,  it  is  evident, 
from  what  has  been  said,  that  the  term — 25  a*  b  must  have 
arisen  from  the  multiplication  of  5(^  by  the  term  affected  with 
the*  highest  exponent  of  the  letter  a  in  the  remaining  terms  of 
the  quotient  sought;  whence," dividing  —  25a^b  by  5a^,  we  shall 
be  sure  to  obtain  a  new  term  of  the  quotient. 

"With  regard  to  the  sign,  which  should  be  prefixed  to  this  term 
of  the  quotient,  it  is  evident,  that  it  should  be  the  sign  — ;  since, 
from  the  nature  of  multiplication,  the  divisor  having  the  sign  -)-, 
the  quotient  must  have  the  sign  —  in  order  that  their  product 
may  produce  anew  the  dividend  —  25a^b. 

Performing  the  operation  therefore,  we  have — 5ab  for 
another  term  of  the  quotient  sought.  Multiplying  the  divisor 
by  this  term  of  the  quotient,  we  obtain  all  the  terms  of  the 
dividend,  which  depend  upon  —  5ab,  viz. 

—  25aH^-\-20aH''  —  25a'b; 
subtracting   these  from   30  a^  ^^  —  25  a*  ^  +  10  a  &*  —  33  a^  b\  the 
remainder  lOa^i^^-j- lOai*  —  8a^3^  will  contain  all  the  partial 
products   arising  from  the  multiplication   of  each  one   of  the 

D 


38  ELEMENTS    OF   ALGEBRA.   • 

remaining  terms  of  the  quotient  sought  by  each  term  of  the 
divisor;  whence,  for  the  same  reasons  as  before,  dividing  10 a^^* 
by  5a^j  we  have  2b^  for  a  new  term  of  the  quotient;  muhiplying 
the  divisor  by  this  term  and  subtracting  as  before,  nothing  re- 
mains; the  division  is  therefore  exact,  and  we  have  for  the 
quotient  sought  4 a'^  —  5ab-]-2b^. 

33.  In  the  course  of  reasoning  pursued  above,  we  have  been 
obliged  to  seek  in  each  of  the  partial  operations  the  term  in 
the  dividend  affected  with  the  highest  exponent  of  one  of  the 
letters,  in  order  to  divide  it  by  the  term  of  the  divisor,  affected 
with  the  highest  exponent  of  the  same  letter.  We  avoid  this 
research  by  arranging  the  dividend  and  divisor  with  reference 
to  the  same  letter;  for,  by  means  of  this  preparation,  the  first 
term  at  the  left  of  the  dividend  and  the  first  term  at  the  left  of 
the  divisor  will,  in  each  of  the  partial  operations,  be  the  two 
terms  which  must  be  divided,  one  by  the  other,  in  order  to  obtain 
a  term  of  the  quotient. 

The  following  is  a  table  of  the  calculations  in  the  preceding 
example,  the  dividend  and  divisor  being  arranged  with  refer- 
ence to  the  letter  a,  and  placed  one  by  the  side  of  the  other  as 
in  arithmetic. 


20a^  — 4U*H 

-50an''  —  Z'iaH^-^\0ab' 
-  20  aH'' 

5a'- 

-.4.aH  +  5ab^ 

20a'—l6a'b-\ 

^a"- 

-5ab'\-2b^ 

—  25c'h- 

—  25a-u- 

-20aH^  —  25aH^ 

-\Oab' 
-  lOab"^ 

From  what  has  been  done  we  have  the  following  rule  for  the 
division  of  compound  quantities,  viz. 

Having  arranged  the  divisor  and  dividend  with  reference  to 
the  powers  of  the  same  letter,  P.  Divide  the  first  term  of  the 
dividend  by  the  first  term  of  the  divisor,  the  result  will  be  the  first 
term  of  the  quotient ;  2°.  multiply  the  whole  divisor  by  the  term 
of  the  quotient  just  found,  and  subtract  the  result  from  the  divi- 
dend; 3°.  divide  the  first  term  of  the  remainder  by  the  first  term 


DIVISION   OF   ALGEBRAIC    QUANTITIES.  39 

of  the  divisor i  the  result  will  be  the  second  term  of  the  quotient ; 
4°.  multiply  the  lohole  divisor  by  the  second  term  of  the  quotient, 
and  subtract  the  product  from  the  result  of  the  first  operation, 
and  continue  the  same  course  of  operations  until  all  the  terms  of 
the  dividend  are  exhausted. 

Recollecting,  that  in  multiplication  the  product  of  two  terms 
affected  with  the  same  sign  should  have  the  sign  -[-»  and  that 
the  product  of  two  terms  affected  with  different  signs  should 
have  the  sign  — ,  we  infer  1°.  that  if  the  two  terms  of  the  divi' 
dend  and  divisor  have  each  the  same  sign,  the  quotient  arising 
from  their  division  should  have  the  sign  -\- ;  but  if  they  are 
affected  with  co7itrary  signs  it  should  have  the  sign  — .*  This  is 
the  rule  for  the  signs. 

EXAMPLES. 

To  divide  « 

by  a^\-Ua\ 

2.  a:^ +  6:c^4- 92-^  4-9ar^  + 4x4-1 

by  ar^4-:c+l.  .  ^ 

3.  ^2x'  —  IQx^y  —l()xhf  -\-\l  x]^  -{'^y'^ 
by  I2x'  —  5xy  —  ^y'. 

4.  x'  —  x^y  —  132-y  +  x'f  -J-  \2xy'' 
by  y^J^^x'y  —  ^xf. 

34.  The  dividend  and  divisor  heing  arranged  with  reference 
to  the  powers  of  the  same  letter,  if  the  first  term  of  the  divi- 
dend is  not  divisible  by  the  first  term  of  the  divisor,  we  infer 
that  the  total  division  is  impossible,  or  in  other  words,  that 
there  is  no  polynominal,  which  multiplied  by  the  divisor  will 
reproduce  the  dividend  ;  and,  in  general,  we  infer  that  the  divis- 
ion cannot  be  exactly  performed,  when  the  first  term  of  any  one 
of  the  partial  dividends  is  not  divisible  by  the  first  term  of  the 
divisor. 

When  the  division  cannot  be  exactly  performed,  in  order  to 
complete  the  quotient,  we  write  the  remainder  over  the  divisor 


40  ELEMENTS    OF   ALGEBRA. 

in  the  form  of  a  fraction  and  annex  it  to   the  quotient  as  in 
arithmetic. 

EXAMPLE. 

To  divide 

hy  5a*  — 2an  +  4:a''b; 
Answer  a^  —  4:a^b'\-2b^ 


5a'  —  2a'b-\-4^d'b^ 
35.  We  may  remark  in  passing,  that  there  is  some  analogy 
between  division  in  arithmetic  and  division  in  algebra  with  re- 
gard to  the  manner  in  which  the  calculations  are  disposed  and 
performed;  there  is,  however,  this  essential  difference,  that  in 
arithmetical  division  the  iigures  of  the  quotient  are  obtained  by 
trial;  whereas,  in  algebraic  division,  we  obtain  with  certainty  a 
term  of  the  quotient  sought,  by  dividing  the  first  term  of  each 
partial  dividend  by  the  first  term  of  the  divisor.  In  Algebraic 
division,  moreover,  we  may  begin,  as  it  will  be  easy  to  see  from 
the  remarks,  art.  26,  at  the  right  instead  of  the  left  of  the  divi- 
dend, since,  in  this  case,  we  shall  have  merely  to  operate  upon 
the  terms  affected  with  the  lowest,  instead  of  those  affected  with 
the  highest  exponents  of  the  letter,  in  reference  to  which  the 
arrangement  is  made ;  whereas,  in  arithmetical  division,  we 
must  always  begin  at  the  left.  Indeed,  such  is  the  indepen- 
dence of  the  partial  operations  in  algebraic  division,  that  having 
obtained  one  of  the  terms  of  the  quotient  and  subtracted  from 
the  dividend  the  product  of  this  term  by  the  divisor,  we  may  in 
the  second  partial  operation  divide,  one  by  the  other,  the  two 
terms  of  the  new  dividend  and  the  divisor  affected  with  the 
highest  exponent  of  any  other  letter  different  from  that,  with 
reference  to  which  the  arrangement  is  made,  and  thus  obtain  a 
new  term  of  the  quotient.  It  is  indeed  only  for  the  sake  of 
convenience,  that  we  always  regard  the  same  letter  in  the  course 
of  the  partial  operations  necessary  to  obtain  the  quotient. 


DIVISION   OF   ALGEBRAIC    QUANTITIES.  41 

36.  In  the  process  of  division,  the  muhiplication  of  the  differ* 
ent  terms  of  the  quotient  by  the  divisor  often  produces  terms, 
which  are  not  found  in  the  dividend,  and  which  it  is  necessary 
to  divide  by  the  first  term  of  the  divisor.  These  terms  are  such, 
as  cancel  each  other  in  the  process  of  forming  the  dividend  by 
the  muhiplication  of  the  divisor  by  the  quotient. 

To  divide  c^  —  J^  by  a  —  b, 


c^  —  b' 
a'  —  aH 

a  —  b 

d^J^ab-^b^ 

aH  —  P 
a^b  —  ab^ 

ab'  —  b' 
ab'  —  b' 

If  we  now  multiply  the  divisor  by  the  quotient  in  this  exam- 
ple, in  order  to  produce  anew  the  dividend,  we  shall  find,  that 
the  new,  terms,  which  arise  in  the  process  of  division,  are  those 
which  cancel  each  other  in  the  result  of  multiplication. 


EXAMPLES. 


1.  To  divide  6a;*  —  96  by  3a:  —  6. 

2.  To  divide  a;'  -j-  ^  by  a:  +  !/• 

3.  To  divide  a*  —  a;*  by  fl  —  x. 

4.  Todividea:'  — a:*  +  a:^  — ar'  +  2a:  — Ibyar^  +  a;— 1. 

37.  It  sometimes  happens,  that  one  or  both  of  the  quantities, 
proposed  for  division,  contains  several  terms  affected  with  the 
same  power  of  the  letter,  in  reference  to  which  the  arrangement 
is  made.  The  following  examples  will  exhibit  the  course  to  be 
pursued  in  cases  of  this  kind. 

1.  To  divide 
Ua'b  —  19abc+  lOa'  —  15a«c  -f  Sab^-{-  ISbc"  —  5Pc    by 
5fl'  +  3ai  — 53c. 

The    terms   11  a^b  —  I5a^c    may  be    put    under    the  form 

(113  —  15  c)  a^j  or  which  is  the  more  convenient  method 

lib 
—  15c 


42 


ELEMENTS    OF   ALGEBRA. 


a  vertical  line  being  employed  instead  of  a  parenthesis  to  in- 
dicate that  the  quantities  lib,  —  15c,  placed  one  under  the 
other  at  the  left  hand,  are  multiplied  each  by  a^.  In  like 
manner   the   terms   — 19abc-\-3ab^  may  be    put  under   the 

form  —  195  c    a 
+    3b^ 

Arranging  the  quantities  with  reference  to  the  letter  a,  the 

calculations  may  be  performed  as  follows. 

5f+2ab--5bc 


lOa'  +  llb 

a'—ldbc 

a  —  dPc+ldbc" 

—  15c 

+  3b' 

10a' -{-   6b 

a'—lObc 

a 

2a  +  b 

Q 


HC 


1st  Rem. 

5b 

—  15c 

5b 

—  15c 


a'  —  dbc 
-\-3b^ 

a'  —  dbc 
+  3P 

a  —  5b^c-\-15bc' 
a  —  5b'c-{-15bc' 


2d  Rem.  0 

Dividing  first  10 a^  by  5a^,  we  have  2a  for  the  quotient; 
subtracting  the  product  of  the  divisor  by  2  a  from  the  dividend, 
we  obtain  the  first  remainder ;  dividing  the  part  affected  with  a' 
in  this  remainder  by  5a^,  we  obtain  b  —  3c  for  the  quotient; 
multiplying  successively  each  term  in  the  divisor  by  3  —  3  c,  we 
exhaust  the  dividend ;  whence  the  quotient  is 
2a-\-b  —  3c. 

In  like  manner  the  following  examples  may  be  performed. 
2.  To  divide 


a^  —  V" 
+  2c2 


a'-\-b' 


b' 

2b'(? 
-\-b''c' 


bya2_^2_g2 


Answer  —  a*  - 

-24' 

a 

''  —  b' 

■v<? 

—  v'e 

3.  To  divide 

• 

f    :^  —  iyz3?  —  'if 

x  +  y' 

-^                  +^yz 

—  1? 

Answer        y 

x'  +  y 

x  —  y 

+' 

^ 

+  ^ 

by 


y\x  —  y 
■z  I     — z 


DIVISION    OF    ALGEBRAIC    QUANTITIES. 


43 


38.  When  the  dividend  is  not  divisible  by  the  divisor,  we  may 
still  attempt  the  division,  according  to  the  rules  which  have  been 
given,  and  continue  it  at  pleasure. 

Thus  let  it  be  required  to  divide  xhy  x-\'  z. 


X 

x-^-z 


X'\-  z 

a:   '    ar      x^  ^   x^ 


—  z 


^ 

X 

x^x' 

z' 

■"? 

2' 

z' 

x'" 

-? 

z'  , 

From  the  number  of  terms  in  the  quotient  already  obtained  in 
the  above  example,  the  learner  will  readily  infer  a  law,  by 
which  the  quotient  may  be  continued  at  pleasure  without  per- 
forming any  more  operations. 

39.  Miscellaneous  examples  in  the  division  of  algebraic  quan- 
tities. 

1.  To  divide  ar'  +  1  by  a:  +  1. 

2.  Todivide  1  — 5a: +10ar^—10a;^  +  5a;^--a:« 
byl  — 2a;  +  ar». 

3.  To  divide  a^  -^  ct^^  hy  a -\-x. 

4.  To  divide  a' -- 5  a' z -\- 10  a^  z""  —  lOa'z' -\- 5az* —  z" 
by  a^  —  2az-\-  z^. 

5.  To  divide  6x'  —  5x'f  +  2l3^y^  —  6x*y*  +  x'f+l5'/ 
hy2:i^  —  3x'f  +  5y\ 


44  ELEMENTS    OF   ALGEBRA. 

6.  To  divide  1  by  1  —  «. 

7.  To  divide  ahj  a  —  ^. 

8.  To  divide  I  ^ax -\-bx'-\' C3^ -{-dx*  +  &c., 
by  1  —  X. 

Ans.  1  4-  1 1  a:  4-  1 1  a;*^  &c. 


9.  To  divide  a  —  bx-\-cx^  —  da^  -{-  &c.,  by  1  -f-  a:. 


SECTION  IV.— Algebraic  Fractions. 

40.  When  the  division  of  two  algebraic  quantities  cannot  be 
exactly  performed,  the  quotient,  as  we  have  seen,  is  expressed  in 
the  form  of  a  fraction,  the  dividend  being  taken  for  the  numera- 
tor and  the  divisor  for  the  denominator. 

A  fraction  in  algebra  has  the  same  signification  as  a  fraction 
in  arithmetic ;  the  denominator  shows  into  how  many  parts  unity 
is  divided,  and  the  numerator  how  many  of  these  parts  are 
taken.  Thus,  in  the  algebraic  fraction  -,  unity  is  supposed  to  be 
divided  into  b  parts,  and  a  number  a  of  these  parts  is  supposed 
to  be  taken. 

reduction  of  fractions  to  their  lowest  terms. 

41.  A  fraction  is  said  to  be  in  its  lowest  terms,  when  there  is 
no  quantity,  that  will  divide  both  of  its  terms  without  a  remain- 
der. To  reduce  a  fraction  therefore  to  this  state,  we  suppress  in 
the  numerator  and  denominator  the  factors,  which  are  common 
to  them. 

The  suppression  of  a  factor  is  the  same,  it  is  evident,  as  divid- 
ing by  the  factor,  required  to  be  suppressed. 

When  the  two  terms  of  an  algebraic  fraction  are  simple  quan- 
tities, it  will  be  easy,  from  inspection,  to  determine  the  factors 


ALGEBRAIC    FRACTIONS.  45' 

common  to  them ;  but  if  the  terms  of  the  fraction  are  polynomi- 
als, this  will  not  be  so  easy,  and  we  must  in  this  case  have 
recourse  to  the  method  of  the  greatest  common  divisor. 

By  the  greatest  common  divisor  of  two  algebraic  quantities 
we  understand  the  greatest  in  regard  to  coefficients  and  expo- 
nents, that  ivill  exactly  divide  these  quantities.  Its  theory  rests 
upon  the  same  two  principles,  as  that  of  the  greatest  common 
divisor  in  arithmetic,  viz. 

1°.  The  greatest  divisor  common  to  tivo  quantities  contains  as 
factors  all  the  particular  divisors  common  to  these  quantities  and 
does  not  contain  any  other  factors.  2°.  The  greatest  divisor 
common  to  two  quantities  is  the  same  with  the  greatest  divisor 
common  to  the  less  of  these  quantities  and  the  remainder  after 
the  division  of  the  greater  by  the  less. 

42.  A  quantity  is  said  to  be  prime  in  respect  to  another  quan- 
tity, when  the  two  have  no  factor  in  common. 

From  the  first  of  the  preceding  principles  it  follows  that  we 
may  multiply,  or  divide,  either  of  the  two  quantities  by  any 
quantity  which  is  prime  to  the  other,  without  affecting  their 
greatest  common  divisor ;  for  we  shall  not,  by  these  operations, 
either  introduce  or  throw  out  a  factor  common  to  the  two 
quantities;  the  greatest  common  divisor,  therefore,  should  re- 
main the  same. 

This  being  premised,  let  it  be  proposed  to  find  the  greatest 
common  divisor  of  the"  polynomials 

(^^aH-\-^ab''—^b\  and  a^  —  5ab-\-U\ 

Pursuing  the  same  general  course  as  in  arithmetic,  we  com- 
mence by  dividing  the  first  of  the  proposed  polynomials  by  the 
second;  we  thus  obtain  a-\-^b  for  a  quotient,  with  a  remainder 
lQab^—\^b\ 

By  the  second  of  the  above  principles  the  question  is  now 
reduced  to  finding  the  greatest  common  divisor  to  this  remainder 
and  the  divisor  a^^5ab-\'W'.  But  19a^2— 19i'  may  be 
put  under  the  form  19  5^  (a  —  h)\  and  since  the  factor  193'^of 


46  ELEMENTS    OF    ALGEBRA. 

this  quantity  is  prime  to  a^  —  5ab-{-4Lp,  it  may,  in  virtue  of  the 
first  of  the  above  principles,  be  suppressed;  thus  the  question 
will  be  still  further  reduced  to  finding  the  greatest  common 
divisor  to  a  —  b  and  a^  —  5a3-|-4Z»^ 

Dividing  the  last  of  these  two  quantities  by  the  first  we  obtain 
an  exact  quotient  a  —  43;  whence  a  —  bis  their  greatest  com- 
mon divisor;  and  by  consequence  it  is  the  greatest  common 
divisor  of  the  polynomials  proposed. 

The  following  is  a  table  of  the  calculations. 
1st  operation  a^>—a'b-}-2ab''^3b^    )  a^  —  5ab  +  W^ 


/^d'b  —  ah'  —  3  // 
/^aH  —  2^ab''+\Qb^ 

or  l^b''{a  —  b) 

a  —  b 


2d  operation  c^  —  5ab-\-^h 
c^  —  ab 


a  —  4:b 


0 

2.  To  find  the  greatest  common  divisor  of  the  polynomials 

a^^5a'b  +  3ab''  —  b'\  .  ,    , 

a^J^2ab4b'  1  Ans.a  +  b. 

3.  To  find  the  greatest  common  divisor  of  the  polynomials 

4.  To  find  the  greatest  common  divisor  of  the  polynomials 

a^  +  ^x^+Sf  ^  ^    1  Ans.  :.  +  3y. 

5.  Let  it  be  proposed,  next,  to  find  the  greatest  common  divisor 
of  the  polynomials 

5b'—lSb^a  +  nba'--6a\sind7b^-^2Sba+6a^ 

In  this  example  5b^,  the  fijst  term  of  the  dividend,  is  not 

divisible  by  7  b^,  the  first  term  of  the  divisor.     It  will  be  observed, 

however,  that  7,  the  coefiicient  of  the  first  term  of  the  divisor, 

will  not  divide  the  remaining  terms  of  the  divisor.        We  may, 


ALGEBRAIC    FRACTIONS.  47 

therefore,  in  virtue  of  the  first  principle,  muhiply  the  dividend 
by  7  without  affecting  the  greatest  common  divisor  sought.     Per- 
forming this  operation,  we  have  for  the  dividend 
S5b^—126b^a  +  77ba'''-A2a\ 

Dividing  next  35^^  by  7b'^,  we  obtain  53  for  a  quotient.  Mul- 
tiplying the  whole  divisor  by  db,  and  subtracting,  we  have  for  a 
remainder  —  11  b"^  a -^  47  b  a^  —  42  a^ 

The  exponent  of  b  in  this  remainder,  being  equal  to  the  expo- 
nent of  the  same  letter  in  the  divisor,  we  continue  the  operation; 
and  in  order  to  render  the  first  term  divisible  by  the  first  term 
of  the  divisor,  we  multiply  anew  by  7,  which  gives  —  77  b^  a 
-{- 329 ba^  —  294 a^.  Dividing  this  by  the  divisor,  the  quotient 
is  —  11a,  which  we  separate  froni  the  other  by  a  comma,  to  show 
that  it  has  no  connection  with  it,  and  the  remainder  is  76  ba^ 
—  228a^or76a2(3  — 3  a). 

Suppressing  the  factor  76  a*^,  the  question  is  reduced  to  finding 
the  greatest  divisor  common  to  b  —  3  a  and  7b^  —  23  3  a  -|-  6  a^. 
Dividing,  therefore,  the  last  of  these  quantities  by  the  first,  we 
obtain  an  exact  quotient  7b  —  2a;  whence  b  —  3a  is  the  greatest 
common  divisor  sought. 

See  a  table  of  the  calculations. 
1st  operation 

25b'— 126Pa -^77 ba''  —  42a^  I  7b^  —  23ba  +  6a' 
35b'—115b^a-]-20ba'  \  qJ^ —11a 


2d  operation 


— 

-lib' a 

-_|-47^^2_ 

42  a^ 

— 

-77  b'a 

',-\-329ba'- 

-294  a' 

— 

-77b''a-\-2d3bd'- 

-66  a' 

7Ud'  — 

■22Sa^ 

or 

7Qa^b- 

-3a) 

>n 

7P- 
7b'_- 

—  23b. 

—  21b 

—  2ha 

a  +  Qd?)^     b 
+  6a'' 

—  3a 

—  2o 

—  2ba 

+  6a'^ 

48  ELEMENTS    OF    ALGEBRA. 

6.  To  find  the  greatest  common  divisor  of  the  polynomials 

7.  To  find  the  greatest  common  divisor  of  the  polynomials 

4:r^_5:.^  +  /        ^J  Ans.  a:-y. 

The  suppression  of  a  factor  common  to  all  the  terms  in  the 
first  remainder  in  the  preceding  examples,  serves  not  only  to 
simplify  the  calculations,  but  is  also  indispemable.  Looking  at 
the  first  example,  it  is  evident  that  unless  the  factor  lO^'^in 
the  first  remainder  be  suppressed,  we  must  multiply  all  the 
terms  of  the  new  dividend  by  19  3*^,  in  order  to  render  the  first 
term  divisible  by  the  first  term  of  the  divisor;  we  should  thus 
mtroducft  into  the  dividend  a  factor,  which  is  also  contained  in 
the  divisor,  and  by  consequence  we  should  introduce  into  the 
greatest  common  divisor  sought,  a  factor,  which  does  not  belong 
to  it. 

8.  Let  it  be  proposed  next  to  find  the  greatest  common  divisor 
of  the  polynomials 

\6a'  -{-lOa'b  +  Mb''  -\-Q>aH^  ^^ab' 
\2aH''  +  2QaH^  +  lQab'—l(ib\ 
Before  proceeding  to  the  division  of  the  proposed  polynomials, 
we  observe  that  the  first  contains  the  letter  a  as  a  factor  common 
to  all  its  terms ;  and  since  this  letter  does  not  enter  as  a  factor 
into  the  second  polynomial,  we  may  suppress  it,  as  forming  no 
part  of  the  greatest  common  divisor  sought. 

For  a  similar  reason,  the  factor  2^^  may  be  suppressed  in  the 
second  polynomial.  Thus  the  question  is  reduced  to  finding  the 
greatest  common  divisor  of  the  polynomials 

\5a'  +  lOa^i  +  Mb^  +  6  ab^^^h" 
^a^+l^aH^-Sab^  —  ^bK 
Pursuing  with  these  polynomials  the  same  course  as  in  the 
preceding  examples,  we   should  multiply  the  dividend  by  G, 


ALGEBRAIC    FRACTIONS.  49 

the  coefficient  of  the  first  term  of  the  divisor.  But  since  15  and 
6  have  a  common  factor  3,  it  will  he  sufficient  to  multiply  hy  2 
the  other  factor  of  6,  which  does  not  enter  into  15 ;  multiplying 
therefore  by  2  and  continuing  the  operations  as  above,  we  obtain 
for  the  greatest  common  divisor,  Sa'^  -|-  2a5  —  b"^. 

9.  To  find  the  greatest  common  divisor  of  the  polynomials 

c  3       /^  2       o         1       '       t  Ans.   2x — 1. 

10.  To  find  the  greatest  common  divisor  of  the  polynomials 

^x^y5a^x^  +  2Wx    1  Ans.  2.:  +  3«. 

From  what  has  been  done,  we  have  the  following  rule,  by 
which  to  find  the  greatest  common  divisor  of  two  polynomials, 
viz.  The  polynomials  proposed  being  arranged  with  reference 
to  the  same  letter,  P.  We  suppress  in  each  the  monomial  factors 
which  are  not  found  in  the  other  ;  2°.  we  divide  one  of  the  poly- 
nomials by  the  other,  and  if  the  division  cannot  be  exactly 
performed,  we  divide  the  first  divisor  by  the  remainder,  and  so 
on,  observing  to  prepare  each  dividend  when  necessary  in  such  a 
manner,  as  to  render  the  first  term  divisible  by  the  first  term  of 
the  divisor,  and  to  suppress  in  each  remainder  the  monomial  fac- 
tors, which  are  n/)t  contained  in  the  preceding  divisor ;  and  that 
remainder,  which  will  exactly  divide  the  preceding,  will  be  the 
greatest  common  divisor  sought. 

43.  The  research  for  the  greatest  common  divisor  of  two 
polynomials  admits,  in  certain  cases,  of  simplifications  which  we 
shall  now  explain. 

1.  Let  it  be  proposed  to  find  the  greatest  common  divisor  of 
the  polynomials 

5a«  +  10a^a:  +  5aV 
a^x-\-2a^x'+2a^x^-\-ax*. 
The  letter  a,  it  will  be  perceived,  enters  as  a  factor  into  each 
of  the  terms  of  the  polynomials  proposed.     This  letter  will, 
therefore,  be  a  factor  of  the  greatest  common  divisor  sought. 
4 


50 


ELEMENTS    OF   ALGEBRA. 


Suppressing  a  in  the  proposed,  and  applying  the  rule  to  the  poly- 
nomials which  result,  we  obtain  a  -[-  a;  for  their  greatest  common 
divisor.  '  The  greatest  common  divisor  sought  will,  therefore,  be 
a{a-\-  a;),  or  c^ -\-  ax. 

2.  To  find  the  greatest  common  divisor  of  the  polynomials 

3.  To  find  the  greatest  common  divisor  of  the  polynomials 

6a*3  —  10  aH^  +  7  aH"^  —  Sab' 
Sa'b  —  5a^b^  +  2ab' 

4.  Let  it  be  required  next  to  find  the  greatest  common  divisor 
of  the  polynomials 


>        Ans.  ab  {a  —  b). 


a'  +  b' 

—  b(? 

a'-^b"" 

—  be 


«5  +  ^V 

—  b^c' 

a^-^-b' 

—  b-'c 


cS- 


The  proposed,  it  will  readily  be  perceived,  have  a  simple  factor 
c^  common  to  both ;  recollecting  that  this  will  be  a  factor  of  the 
greatest  common  divisor  sought,  we  suppress  it,  and  the  polyno- 
mials, which  result,  will  be 


a'  +  W 
—  b& 


a^-\-b^ 
—  be 


a^b\ 
—  bH 


We  may  now  commence  the  division  of  one  of  these  polyno- 
mials by  the  other  according  to  the  rule,  in  order  to  determine 
their  greatest  common  divisor.  Before  proceeding  to  this,  how- 
ever, let  us  see  if  there  be  not  a  polynomial  divisor  common  to 
the  coefficients  of  the  letter  a,  with  reference  to  which  the 
arrangement  is  made. 

Comparing  for  this  purpose  the  two  coefficients  of  the  lowest 
degree  b'^  —  (?  and  b  —  c,  we  find  that  b  —  c  will  divide  both 
without  a  remainder.  We  inquire  next  if  i  —  c  will  divide  the 
remaining  coefficients  of  a.  This  is  the  case;  b  —  c,  therefore, 
is  a  divisor  common  to  all  the  coefficients  of  the  two  last  polyno- 
mials.    Recollecting  that  b  —  c  will  also  be  a  factor  of  the 


ALGEBRAIC    FRACTIONS. 


51 


greatest  common  divisor  sought,  we  suppress  b  —  c,  and  the 
polynomials,  which  result,  will  be 


b 


"t 


^3  ^  33.^2  _|.  32^^  and  a^  +  3a  +  b^ 


Applying  the  rule  to  these,  the  first,  it  will  be  perceived, 
contains  a  factor  b-^Cj  which  is  not  contained  in  the  second. 
Suppressing  this,  it  remains  to  find  the  greatest  common  divisor 
of  the  polynomials 

a*  +  ba'  +  b'c\  and  ^=  +  *a  +  b\ 
These,  it  will  be  found,  have  no  common  divisor.     The  greatest 
common  divisor  of  the  proposed  will,  therefore,  be 
a^  {b  —  c),  or  a^b  —  a^c. 
7.  To  find  the  greatest  common  divisor  of  the  polynomials 

y 


X 

-1 

2/5  — 3a; 
+  3 

+  3a; 
—  2 

y'  +  c^ 

X 

1 

+  3 

f-\-x\ 

X 

—  2 

+  x 

y- 

Ans.  y{y—\)  {x^l). 

8.  Let  it  be  proposed  next  to  find  the  greatest  common  divisor 
of  the  polynomials 


—  yz' 


fz 
yz^ 


x'  +  by' 

—  byz 

—  cyz"  I 

x^  -\-by^z 
--dy^z 

—  by^ 

—  dyz" 


:j^-\-bc'f 
—  bcyn^ 


x-\-bd'ifz 
—  bdy^ 


The  simple  quantity  xy^  it  will  be  perceived,  will  exactly 
divide  each  of  the  terms  of  the  first  of  the  proposed  polynomials, 
and  yz  those  of  the  second.  The  factor  y  common  to  these 
quantities  will  be,  it  is  evident,  a  factor  of  the  greatest  common 
divisor  sought.     Setting  apart  the  y  therefore  as  such,   and 


S^  ELEMENTS    OF    AteEBEA. 

dividing  the  first  polynomial  hy  xy  and  the  second  by  yzy  Uie 
polynomials,  which  result,  will  be 


—  bcz" 


x-^hdy 
—  bdz 


y\x^-{-by 

—  z  I       -{-dy 

—  bz 

— dz 

The  coefficients  of  the  first  of  these  aire  divisible  each  by 
^  —  z^  and  those  of  the  second  by  y  —  z;  but  y  —  2,  being 
a  factor  common  to  y^  —  :^  and  y  —  z,  will  also,  it  is  evident, 
be  a  factor  of  the  greatest  common  divisor  sought;  setting  it 
apart,  therefore,  as  such,  and  dividing  the  first  polynomial  by 
y*  —  2^  and  the  second  by  2/  —  z,  the  polynomials,  which  result, 
will  be 

-|-c|  \d\ 

Applying  the  rule  to  these  last,  we  obtain  x  -J-  3  for  their 
greatest  common  divisor.  The  greatest  common  divisor  of  the 
proposed  will,  therefore,  be  y{y  —  z)  [x-\-b). 

From  what  has  been  done,  the  following  method  for  finding 
the  greatest  common  divisor  of  two  polynomials  will  be  readily 
inferred,  viz. 

1*.  Suppress  in  the  polynomials  proposed  the  greatest  simple 
divisors,  which  they  respectively  contain,  observing  to  set  aside  as 
a  factor  of  the  greatest  common  divisor  sought,  the  greatest 
factor,  which  these  divisors  have  in  common.  2°.  Suppress  in 
the  polynomials,  which  result,  the  greatest  polynomial  divisor, 
independent  of  the  principal  letter,  and  set  aside  as  a  factor  of 
the  greatest  common  divisor  sought  the  greatest  factor  which 
these  divisors  have  in  common.  3°.  Find  the  greatest  common 
divisor  of  the  polynomials  which  result,  this  will  be  the  remain- 
ing factor  of  the  greatest  common  divisor  sought,  and  the  product 
of  the  several  factors,  thus  obtained,  will  be  the  greatest  commoa 
divisor  sought. 


A](.aSBRAIC    FBAOTIONS.  -|3 

44.  To  reduce  a  fraction  to  its  lowest  terms,  we  divide  the  two 
terms  of  the  fraction  by  their  greatest  common  divisor. 

EXAMPLES. 

1.  Reduce  -= -r-^  to  its  lowest  terms.  Ans.  — -f — . 

^4 x'^ 

%  Reduce  -^—. — 5 5 =  to  its  lowest  terms. 

Ans.  — — . 
a-J-a: 

^    _,   ,       Qa7?-\-a3^ — \2ax      .  .      1    /. 

3.  Reduce -^ ^ to  its  most  simple  form. 

Qax — 8a  ^ 

23?4-2x 
Ans.         J      , 

4.  Reduce  -5 — .   .  ,  ^  » — T    1  ^  to  its  lowest  terms.' 

x^  —  4a:^-j-6ar* — ^x-\-2 

Ans.         "■ 


45.  Algebraic  fractions  being  of  the  same  nature  as  fractions 
in  arithmetic,  the  rules  for  the  fundamental  operations  are  the 
same.  We  shall  merely  subjoin  these  rules,  with  some  exam- 
ples und^  each,  the  results  being  reduced  to  their  lowest  terms. 

MULTIPLICATION   OF    ALGEBRAIC    FRACTIONS. 

Rule. — Multiply  the  numerators  together  for  a  new  numerator, 
and  the  denominators  for  a  new  denominator. 

EXAMPLES. 

1.  Multiply  —5  by  - — 5-  Ans.  -„. 

2.  Multiply  — !-^ 1 —  by  — r  -.  Ans.  •  ^     '"-^. 

cd  —  d*  a-\-o  c  —  d 

e    n/r  u'  1    «'*  +  «a:  ,      a^  —  a^  .        (^4-a^x4-aa* 

3.  Multiply     o        o  by -^.        Ans.   — ?^ -\= — . 


a^_l.i         X 1 

4.  Multiply  3:p, -^^~  and  — t— 7  together. 
2a  a-j-o 


Sa:'  — 3a: 
Ans. 


2a^-^-2ab 


54  ELEMENTS    OF    ALGEBRA. 

6,  Multiply  — .—r-t  r — -2  and  a-{ together. 

^•'    a-{-b    ax-\-x^  'a  —  x    ^ 


Ans.  ^(^i). 


X 
DIVISION   OF   ALGEBRAIC    FRACTIONS. 


Rule. — Invert  the  divisor ^  and  then  proceed  as  in  multiplica- 
tion. 


EXAMPLES. 


1.  Divide  i+*  by  i+|.  Ans.  ^*'. 

X  —  y        a  —  b  ar  —  if 

2.  Divide  ^^  by  ^.  Ans.  -^+^. 

a^  —  x^      ^  a  —  x  a^J^ax-{-3? 

3.  Divide  -2 — r—r^  by '—r-*  Ans.  — ' — . 

a^  —  2bx-\-b^    ^     X  —  b  x 

4.  To  divide  12  by  ^-^i^  —  a.  Ans.              "^^^ 


X  c^-^-ax-^-a^' 

ADDITION    OF   ALGEBRAIC   FRACTIONS. 

Rule. — Reduce  the  fractions  to  a  common  denomincUor  ;  then 
add  the  numerators  together,  and  place  their  sum  over  the  common 
denominator. 

EXAMPLES. 

1.  Add  together  ^-±1  and  ^.  Ans.  %±^. 

x  —  y        x-f-y  a^  —  f 

2.  Add  together  ^\nd^^:=^.  Ans.  ^i^+^V  +  f), 

x-j-y  x  —  y  x-j-y 

3.jj^       ,      a  a  —  3^        .  a^  —  b^  —  ab 
.  Add  together  7, — ,  and ; — ; .     ♦ 

o       cd  bed 

.        acd  —  4tb^4-a^ 

Ans. i— 7—^ • 

bed 

4.  Add  together  - — r-— 5,  —  - — 5-— -„,  and 


{a  +  bY'       {a  +  bf         a-\-b 

i 

la  +  bY     ' 


a^A.ab^  +  b* 
Ans.  — * —       ' 


EQUATIONS    OF    THE    FIRST   DEGREE.  55 

5.  Add  together ^r-r:^,    - — j — ^7" ^  n   v»  and j — . 

•    (a  —  2xy     {a-{-x)  {a  —  2x)  a-\-x 

20ax  —  223^ 


Ans. 


{a-^-x)  {a  —  2xf 

SUBTRACTION    OF   ALGEBRAIC    QUANTITIES. 

Rule. — Reduce  the  fractiom  to  a  common  denominator  ;  then 
place  the  difference  of  their  numerators  over  the  denominator ^  and 
it  will  be  the  difference  required. 

EXAMPLES. 

,    ^         5a:  — 3      ^  ^x-\-2       ,        Sar*— 13a:+l 

1.  From  ;-— -  subtract ^;-.      Ans.  5 — ^ — . 

x-\-\  X —  1  ar —  1 

«   T.  1  ,  1  *  2v 

2.  Ironx  subtract   — ; — .  Ans.     „        „• 

X  —  y  ^-tV  ^  —  2r 

-    _  az         ,  a  —  z  .        2az  —  a^  —  z^ 

3.  From   -:; 5  subtract  — -, — .         Ans. 


a^  —  z^  a-\-z  ^  d^  —  ^ 

"T    ^   '       subtract  ^~~   .  Ans.       ^  ,. 

V  —y  y  y—i 


SECTION  V. — Equations  of  the  First  Degree. 

46.  The  rules  obtained  in  the  preceding  sections,  are  suf- 
ficient for  the  solution  of  all  equations  of  the  first  degree, 
however  complicated.  We  place  below  a  few  examples,  in- 
volving operations  a  little  more  complicated  than  those,  which 
have  been  previously  introduced. 

,     _.         7x  —  S.l5x-{-S       _         31— a:    ,      .    ,    ,, 

1.  Given    — — 1 Y^ —  =  3  a: -^ — ,   to   find   the 

value  of  a:.  Ans.  a;  =  9. 

^     „.  2a:+l       402  — 3a:       _       471  — 6a:      ^      .    , 

2.  Given    -^ 12—  =  ^ 2"'     ''    ^^^ 

the  value  of  x.  Ans.  x  =  72. 


66  ELEMENTS    OF   ALGEBRA. 

3.  Given  — ^ —  =  — -+-  -,  to  find  the  value  of  x. 

36  5a;  — 4    '   4 

Ans.  x  =  S. 

.     ^.  10a:+17        12a:  +  2        5a:  — 4     ,      .    ,     . 

4.  Given    _^-^^_  =  ^-^,    to    find    the 

value  of  z.  Ans.  a:  =  4. 

/        18a:— 19   ,    lla:  +  21       9a:+15     ,      .    ,    . 

5.  Given    _^^  +  _^_  =  _^,    to    find    the 

value  of  X.  Ans.  a:  =  7. 


PROBLEMS     AND     EQUATIONS     OF     THE     FIRST     DEGREE     WITH     TWO 
UNKNOWN    QUANTITIES. 

47.  Most  of  the  questions  we  have  hitherto  considered,  in- 
volve more  than  one  unknown  quantity.  We  have  been  able 
to  solve  them,  however,  by  representing  one  of  the  unknown 
quantities  only  by  a  letter,  since,  by  means  of  this,  it  has  been 
easy,  from  the  conditions  of  the  question,  to  express  the  other 
unknown  quantity.  In  many  questions  the  solution  becomes 
more  simple  by  representing  more  than  one  of  the  unknown 
quantities  by  a  letter,  and  in  complicated  questions,  it  is  fre- 
quently necessary  to  do  this.  ' 

The  question,  art.  1.  viz.  To  divide  the  number  56  into  two 
suck  parts  that  the  greater  may  exceed  the  less  by  12,  presents 
itself  naturally  with  two  unknown  quantities.  Thus,  denoting 
the  less  part  by  x  and  the  greater  by  y,  we  have  by  the  con- 
ditions of  the  question 

x~\"y  =  5Q 
y-.x=\2. 

Deducing  the  value  of  y  from  the  second  equation,  we  have 
y=zx-\-12;  substituting  for  y  in  the  first  equation  its  value 
X  -j-  12,  we  have  x  -\-  x  -\-  12  z=  56,  an  equation,  which  con- 
tains only  one  unknown  quantity,  and  from  which  we  obtain 
a:  =  22. 


EQUATIONS   OF  THE   FIRST   DEGREE.  W 

2.  A  person  has  two  horses  and  a  saddle,  which  of  itself  is 
worth  $  10.  If  the  saddle  be  put  upon  the  first  horse,  his  value 
will  be  twice  the  second;  but  if  the  saddle  be  put  upon  the 
second,  his  value  will  be  three  times  the  first.  What  is  the 
value  of  each  ? 

Let  X  =  the  value  of  the  first  horse,  and  y  that  of  the  second, 
we  have  by  the  question 

a:-[-10  =  2y 
y4-10  =  ^«. 

Deducing  the  value  of  y  from  the  second  of  diese  equations, 
and  substituting  it  for  y  in  the  first,  we  have 

a;-f.l0  =  6a:  — 20; 
whence  a:  =  6.  ' 

Substituting  next  for  x  its  value  6  in  the  second  equaticm, 
we  have  z/  -["  1^  =  1® ' 

whence  2/  =  S* 

The  process  by  which  one  of  the  unknown  quantities  in  an 
equation  is  made  to  disappear,  is  called  elimination.  The 
method  of  eliminating  one  of  the  unknown  quantities,  pursued 
above,  is  called  elimination  by  substitution. 

48.  Since  the  two  members  of  an  equation  are  equal  quanti- 
ties, it  is  evident,  P.  that  we  may  add  two  equations,  Tnember  to 
member,  without  destroying  the  equality ;  2°.  we  may  subtract 
the  members  of  one  equation  from  those  of  another  without  destroy- 
ing the  equality. 

Taking  advantage  of  this  remark,  we  may  frequently  elimi- 
nate one  of  the  unknown  quantities  in  a  more  simple  manner, 
than  by  the  process  of  substitution. 

Let  there  be  proposed,  for  example,  the  equations 
5a;  +  7y  =  43 
lla:  +  9y  =  69. 

If  either  of  the  unknown  quantities  in  these  equations  were 
aflfected  with  the  same  coefficient,  we  might,  it  is  evident, 
eliminate  this  unknown  quantity  by  a  simple  subtraction.     But 


'58  ELEMENTS   OF   ALGEBRA. 

if  the  first  equation  be  multiplied  by  9,  the  coefficient  of  y  in 
the  second,  and  the  second  by  7,  the  coefficient  of  y  in  the 
first,  we  shall  obtain  two  new  equations,  which  may  be  sub- 
stituted for  the  proposed,  and  in  which  the  coefficient  of  y  will 
be  equal,  viz. 

45a; +  63?/ =  387 
77  a: +  63?/ =  483. 
Subtracting  then  the  first  of  these  equations  from  the  second, 
we  have   32  a;  =  96,  from  which  we   obtain  a;  =  3.     Substi- 
tuting this  value  of  z  in  either  of  the  proposed  we  obtain  the 
value  of  y. 

In  like  manner,  if  we  wish  first  to  eliminate  a:,  we  multiply 
the  first  of  the  proposed  equations  by  11,  the  coefficient  of  x 
in  the  second,  and  the  second  by  5,  the  coefficient  of  x  in  the 
first ;  we  thus  obtain  two  new  equations,  which  may  be  substi- 
tuted for  the  proposed,  and  in  which  the  coefficients  of  x  will 
be  equal,  viz. 

55  a: +  77^  =  473 
55  a: +45?/ =  345. 
Subtracting  therefore  the  second  of  these  equations  from  the 
first,  we  have  32?/  =  128 ;  whence  ?/  =  4. 
Let  us  take  as  a  second  example  the  equations 
8a:  — 21?/  =  33 
6a: +  35?/ =  177. 
The  coefficients  of  x  in  these  equations  have,  it  will  be  per- 
ceived, a  common  factor  2.     It  will  be  sufficient  therefore,  in 
order  to  render  these   coefficients   equal,  to  multiply  the  first 
equation  by  3  and  the  second  by  4.     Performing  the  operations 
we  have 

24a:  — 63  2/ =  99 
24a; +140?/ =  708; 
whence,  subtracting  the  first  of  these  equations  from  the  second 

we  obtain 

203y  =  609; 
therefore  2/  =  3. 


EQUATIONS   OF   THE    FIRST   DEGREE.  59 

In  like  manner,  since  the  coefficients  of  y  contain  the  common 
factor  7,  in  order  to  render  the  coefficients  of  y  equal  we  multi- 
ply the  first  of  the  proposed  equations  by  5  and  the  second  by  3, 
which  gives  two  new  equations, 

40  a;— 1052/ =165 
18a: +1052/ =  531; 
whence  by  addition  we  obtain 

58  a:  =  696; 
therefore  x  =  12. 

49.  The  method  of  elimination,  which  we  have  now  ex- 
plained, is  called  elimination  by  addition  and  subtraction,  since, 
the  equations  being  properly  prepared,  we  cause  one  of  the 
unknown  quantities  to  disappear  by  addition  or  subtraction. 

In  the  use  of  this  method,  it  is  important  to  ascertain  whether 
the  coefficients  have  common  factors,  since,  if  this  be  the  case, 
by  omitting  the  common  factors  in  the  multiplications  required, 
the  calculations  to  be  performed  become  more  simple.  The 
equations,  moreover,  should  be  reduced  to  the  form  of  the  pre- 
ceding examples, — that  is,  they  should  be  freed  from  denomi- 
nators, the  unknown  quantities  collected  each  into  one  term  on 
one  side  of  the  sign  of  equality,  and  the  known  quantities  col- 
lected in  one  term  on  the  other. 

EXAMPLES. 

1.  To  find  the  values  of  x  and  y  in  the  equations 

4a:  — 3t/=l 
3a:-j-42/  =  ^. 

2.  To  find  the  values  of  x  and  y  in  the  equations 

4a:  — 9?/  =  51 
8a: +  132/ =  191. 
3»  To  find  the  values  of  x  and  y  in  the  equations 

82/  — 3a:  =  29 

67y  — 4a;  =  20 


•0  »I,EM(ENTS   Q?  AW31SBRA. 

4.  To  find  the  values  of  ^  and  y  in  the  equatious 

Ans.  x=l2,y=il6, 

5.  To  find  the  values  of  x  and  y  in  the  equations 

a;  +  2 


3 

y  +  5 


82/  =  31 


^      +  10a;  =192. 
4       ' 


Ana.  a:=3sI9,  ysBB^k 

6.  To  find  the  values  of  x  and  y  in  the  equations 

2^  +  32,-4=15 

Ans.  3;  =  7, 1/  =  5. 

7.  To  find  the  values  of  x  and  y  in  the  equations 

Ans.  a;  =  3,  y  =  2, 

8.  To  find  the  values  of  x  and  y  in  the  equations 

37/  +  4a:_         9y  +  33 
'*'"+"^  7        ~  14 

„       5a:  — 4y  llw— 19 

Ans.  a;  5=?  6>  2^  ==  5. 

50.  We  pass  next  to  the  solution  of  some  questions  producing 
equations  involving  two  unknown  quantities. 

1.  A  number  consisting  of  two  figures  when  divided  by  4, 
gives  a  certain  quotient,  and  a  remainder  of  3;  when  divided 


eqitatkWS  of  the  first  begree.  61 

by  9  gives  another  qaolient  and  a  remainder  of  8.  The  value 
of  the  figure  on  the  left  hiand  is  equal  to  the  quotient  obtained, 
when  the  number  was  divided  by  9,  and  the  other  figure  is  equal 
to  tV  of  ^^®  quotient  obtained,  when  the  number  was  divided 
by  4.     Required  the  number. 

Let  a:  =  the  figure  in  the  place  of  tens,  y  that  in  the  place 
of  units;  then  10^-{"2/  =  *^6  number,  and  we  have  by  the 
question 

Deducing  the  values  of  x  and  y  from  these  equations,  we 
obtain  a:  ==  7,  y  =  1.     The  number  required  is  therefore  71. 

2.  A  purse  holds  19  crowns  and  6  guineas.  Now  4  crowns 
and  5  guineas  fill  ^^  of  it.     How  many  will  it  hold  of  each  ? 

Let  z  =  the  number  of  crowns  and  y  =  the  number  of 

guineas,  then  ~  =  the  space   occupied  by  a  crown,  and  -  = 

the  space  occupied  by  a  guinea,  we  have  therefore  by  the  ques*- 
tion 

19   ,6       ^        ,4,5       17 

f-  -  =  1,  and  -  +  -  =  ^. 

sc    '   y  z      y       o3 

Multiplying  the  first  equation  by  5  and  the  second  by  6,  subtract- 
ing the  second  from  the  first  and  reducing,  we  obtain  z  =  21, 
whence  y  =  63. 

3.  What  fraction  is  that,  whose  numerator  being  doubled,  and 
denominator  increased  by  7,  the  value  becomes  §j  but  the  de- 
nominator being  doubled,  and  the  numerator  increased  by  2,  the 
value  becomes  f  ?  Ans.  f. 

4.  A  owes  $1200,  B  $2500;  but  neither  has  enough  to  pay 
his  debts.  Lend  me,  said  A  to  B,  the  eighth  part  of  your 
fortune,  and  I  shall  be  enabled  to  pay  my  debts.  B  answered, 
I  cin  discharge  my  debts,  if  you  will  lend  me  the  9th  part  of 
youjfs.    What  was  the  fortune  of  each? 

Ans.  A's  $900,  B's  $2400. 

F 


62  ELEMENTS   OF   ALGEBRA. 

5.  A  farmer  with  28  bushels  of  barley  at  2s.  M.  per  bushel, 
would  mix  rye  at  3  shillings  per  bushel,  and  wheat  at  4  shillings 
per  bushel,  so  that  the  whole  mixture  may  consist  of  100  bushels, 
and  be  worth  35.  4^.  per  bushel.  How  many  bushels  of  rye,  and 
how  many  of  wheat  must  he  mix  with  the  barley  ? 

Ans.  20  bushels  of  rye  and  52  bushels  of  wheat. 

6.  A  and  B  speculate  with  different  sums;  A  gains  $150, 
B  loses  $50,  and  now  A's  stock  is  to  B's  as  3  to  2.  But  if  A 
had  lost  $50  and  B  gained  $100,  then  A's  stock  would  have 
been  to  B's  as  5  to  9.     What  was  the  stock  of  each  ? 

Ans.  A's  $300  and  B's  $350. 

7.  A  rectangular  bowling  green  having  been  measured,  it 
was  observed,  that  if  it  were  5  feet  broader,  and  4  feet  longer,  it 
would  contain  116  feet  more;  but  if  it  were  4  feet  broader,  and 
5  feet  longer,  it  would  contain  113  feet  more.  Required  the 
length  and  breadth. 

Let  a;  =  the  length,  ?/  =  the  breadth,  then  xy=zth.e  content, 
and  by  the  first  condition  (x  -|-  4)  {y  -\-  5)  =  xy  -^  116,  &c. 
Ans.  The  length  was  12  and  the  breadth  9  feet. 

8.  There  is  a  number  consisting  of  two  figures,  the  figure  in 
the  place  of  units  being  the  greater ;  if  the  number  be  divided  by 
the  sum  of  its  figures,  the  quotient  is  4;  but  if  the  figures  be 
inverted,  and  the  number  which  results  be  divided  by  a  number 
greater  by  2  than  the  difference  of  the  figures,  the  quotient 
becomes  14.     Required  the  number.  Ans.  48. 

9.  A  person  has  two  horses  and  two  saddles,  one  of  which 
cost  $50,  the  other  $2.  If  he  places  the  best  upon  the  first 
horse,  and  the  worst  upon  the  second,  then  the  latter  is  worth 
$8  less  than  the  other;  but  if  he  puts  the  worst  saddle  upon 
the  first,  and  the  best  upon  the  second  horse,  then  the  latter  is 
worth  3f  times  as  much  as  the  former.  What  is  the  value  of 
each  horse  ?  Ans.  The  first  $30,  the  second  $70. 

10.  A  cistern  containing  210  buckets,  may  be  filled  by  2 


EQUATIONS    OF    THE    FIRST   DEGREE.  63 

pipes.  By  an  experiment,  in  which  the  first  was  open  4  and 
the  second  5  hours,  90  buckets  of  water  were  obtained.  By 
another  experiment,  when  the  first  was  open  7,  and  the  other  3^ 
hours,  126  buckets  were  obtained.  How  many  buckets  does 
each  pipe  discharge  in  an  hour  ? 

Ans.  The  first  15,  and  the  second  6  buckets. 

11.  A  person  having  laid  out  a  rectangular  bowling  green, 
observed  that  if  each  side  had  been  4  yards  longer,  the  adjacent 
sides  would  have  been  in  the  proportion  of  5  to  4,  but  if  each 
had  been  4  yards  shorter,  the  proportion  would  have  been  4  to  3. 
What  are  the  lengths  of  the  sides  ?  Ans.  36  and  28  yds. 

12.  A  vintner  has  two  casks  of  wine,  from  the  greater  of 
which  he  draws  15  gallons,  and  from  the  less  11 ;  and  finds  the 
quantities  remaining  in  the  proportion  of  8  to  3.  After  the  casks 
become  half  empty,  he  puts  10  gallons  of  water  into  each,  and 
finds  that  the  quantities  of  liquor  now  in  them  are  as  9  to  5. 
How  many  gallons  will  each  hold  ? 

Ans.  The  larger  79  and  the  smaller  35  gallons. 

13.  Two  persons,  A  and  B,  can  perform  a  piece  of  work  in 
16  days.  They  work  together  for  4  days,  when  A  being  called 
oflf,  B  is  left  to  finish  it,  which  he  does  in  36  days  more.  In 
what  time  would  each  do  it  separately  ? 

Ans.  A  in  24  and  B  in  48  days. 

14.  A  work  is  to  be  printed,  so  that  each  page  may  contain 
a  certain  number  of  lines,  and  each  line  a  certain  number  of 
letters.  If  we  wished  each  page  to  contain  3  lines  more,  and 
each  line  4  letters  more,  then  there  would  be  224  letters  more  in 
each  page ;  but  if  we  wished  to  have  2  lines  less  in  a  page,  and 
3  letters  less  in  each  line,  then  each  page  would  contain  145 
letters  less.  How  many  lines  are  there  in  each  page  ?  and  how 
many  letters  in  each  line  ? 

Ans.  There  are  29  lines  in  a  page  and  32  letters  in  a  line. 

15.  There  is  a  number  consisting  of  two  digits,  which  is 


■9fm  ELEMENTS   OF   ALGEBRA. 

equal  to  four  times  the  sum  of  those  digits ;  and  if  18  be  added 
to  it,  the  digits  will  be  inverted.     What  is  the  number? 

Ans.  24. 

16.  To  find  a  fraction  such,  that  if  3  be  subtracted  from  the 
numerator  and  denominator,  it  is  changed  into  ^^  but  if  5  be 
added  to  the  numerator  and  denominator  it  becomes  J.  What 
is  the  fraction?  Ans.  -f^. 

17.  There  is  a  cistern,  into  which  water  is  admitted  by  three 
cocks,  two  of  which  are  of  exactly  the  same  dimensions.  When 
they  are  all  open,  five-twelfths  of  the  cistern  is  filled  in  four 
hours;  and  if  one  of  the  equal  cocks  be  stopped,  seven-ninths 
of  the  cistern  is  filled  in  ten  and  two-thirds  hours.  In  how 
many  hours  would  each  cock  fill  the  cistern? 

Ans.  Each  of  the  equal  ones  in  32 
hours  and  the  other  in  24. 

18.  A  person  owes  a  certain  sum  to  two  creditors.  At  one 
time  he  pays  them  $53,  giving  to  one  four-elevenths  of  the 
sum  due  to  him,  and  to  the  other  $3  more  than  one-sixth  of 
his  debt  to  him.  At  a  second  time  he  pays  them  $42,  giving 
to  the  first  three-sevenths  of  what  remains  due  to  him,  and  to 
the  other  one-third  of  what  is  due  to  him.  What  were  the 
debts?  Ans.  $121  and  $36. 

PROBLEMS    AND    EQUATIONS    OF    THE    FIRST    DEGREE    WITH    THREE 
OR    MORE    UNKNOWN    QUANTITIES. 

51.  Let  now  the  following  question  be  proposed,  viz. 

There  are  three  persons.  A,  B,  and  C,  whose  ages  are  as 
follows ;  If  from  4  times  A's  age  added  to  5  times  B's  age,  we 
subtract  three  times  C's  age,  the  remainder  will  be  70;  if  from  3 
times  A's  age  we  subtract  4  times  B's  age,  and  to  the  remainder 
add  twice  C's  age,  the  sum  will  be  25;  and  if  twice  A's  age,  3 
times  B's,  and  5  times  C's  age  be  added  together,  the  sum  will 
be  240     What  is  the  age  of  each  ? 


EQUATIONS   OF  THE    FIRST   DEGREE.  0S 

This  question  presents  itself  naturally  with  three  unknown 
quantities.  Thus  denoting  A's  age  by  a:,  B's  age  by  y,  and  C's 
by  Zy  we  have  by  the  question 

4a;  +  5y  — 32r  =  70 
3a;  — 4y  +  2z  =  25 
2a;  +  3y  +  5z  =  240. 

Multiplying  the  first  equation  by  2,  and  the  second  by  3,  and 
adding  the  results,  we  obtain 

17x  —  2y=:2l5. 

Again,  multiplying  the  second  equation  by  5,  and  the  third  by 
2,  and  subtracting,  we  obtain, 

—  11a: +  262/ =  355. 

We  have  now  two  equations  with  two  unknown  quantities  only. 
Deducing  next  the  values  of  x  and  y  from  these,  in  the  same 
manner  as  in  the  preceding  equations  with  two  unknown  quanti- 
ties, we  have  a;  =  15,  y  =  20 ;  substituting  these  values  in  the 
first  of  the  proposed  equations,  we  obtain  z  =  30. 

52.  In  the  same  manner,  if  there  be  four  equations,  with  four 
unknown  quantities,  we  combine  the  equations  two  by  two,  until 
one  of  the  unknown  quantities  is  eliminated  from  the  whole; 
we  then  have  three  equations  with  three  unknoAvn  quantities. 
Combining  next  these  three,  two  by  two,  until  one  of  the  un- 
known quantities  is  eliminated,  we  obtain  two  equations  with 
two  unknown  quantities,  and  so  on.  The  process  is  altogether 
similar  for  five  or  more  equations  with  the  same  number  of 
unknown  quantities. 

EXAMPLES. 

1.  To  find  the  values  of  x,  y,  and  z  in  the  equations 
5a;  — 6y  +  42r=15 
7a;-|-4y  — 32=19 
2x-f-     y+62:r=:46. 
,  Ans.  a:  r=  3,  y  s=  4,  z  =  6. 

6 


66  ELEMENTS    OF    ALGEBRA. 

%  To  find  the  values  of  x,  y,  and  z  in  the  equations 
2x-]-4:y  —  3z=^22 
4.x^2y  +  5z=18 
ex  +  7y—    z  =  63. 

Ans.  a;  =  3, 2/  =  7,  z  =  4. 

3.  To  find  the  values  of  x,  y  and  z  in  the  equations 

3x  +  5y  +  7z=179 
Sx-\'3y  —  2z=  64 
5x—    y-\-3z=.   75. 

Ans.  a;  =  8, 2/  =  10,  2  =  15. 

4.  To  find  the  values  of  x,  y  and  z  in  the  equations 

3x-\-2y  —  Az=m  8 
5x  —  3y-\-3z  =  33 
7x+   y  +  5z  =  65. 

Ans.  a;  =  6, 2/  =  3,  2:  =  4. 

5.  To  find  the  values  of  a:,  y,  2r,  and  u  in  the  equations 

a:-(~  y~{~  ^ —  u  =  5 
2x-\-3y  —  2z-^  M==2 
5x—2y-\-  z  —  2u  =  9 
3X+    y  —  2z-\-    u  =  2. 

Ans.  a;  =  2, 2/=l,  z  =  3,  M=l. 
53.  It  sometimes  happens,  that  all  the  unknown  quantities  are 
not  found  in  each  of  the  equations.     In  this  case,  the  elimination 
may,  with  a  little  attention,  he  very  readily  performed. 

1.  Let  there  be  proposed,  for  example,  the   four  following^ 
equations,  with  four  unknown  quantities,  viz. 
2x  —  3y-\-2z=\3 
4w  — 2a:  =  30 
42/  +  2z=14 
5y+ 32^=32. 
With  a  little  examination  we  see,  that  the  elimination  of  z 
from  the  first  and  third  equations  will  give  an  equation  in  z  and 


EQUATIONS   OF  THE    FIRST  DEGREE.  6Jf 

y,  and  that  the  elimination  of  u  from  the  second  and  fourth  equa- 
tions will  also  give  an  equation  in  x  and  y.  From  these  last  the 
values  of  z  and  y  may  be  readily  found.  Performing  the  neces- 
sary operations  we  obtain  a:  =  3,  y  =  1.  Substituting  next  for 
z  its  value  in  the  second  equation,  we  have  w  =  9,  and  substi- 
tuting for  y  its  value  in  the  third,  we  have  2;  =  5. 

2.  To  find  the  values  of  a:,  y,  z  and  u  in  the  following  equa- 
tions. 

3a:—  y  +  2z  =  7 
5x-\-2y —  u==5 
2z—Sy-{-2z  =  2 

7y— 3w  =  2. 

Ans.  a:=l,  y  =  2,  z=3,  M  =  4. 

3.  To  find  the  values  of  the  unknown  quantities  in  the  follow- 
ing equations 

5z—   y  — 3z  =  8 
Sy  —  2z-\-    t  =  0 
z-\-2y—r  z  =  3 
5y —   u-{'5t  =  5 
4?^+   M  =  9. 
Ans.     a:  =  3, 2/=  1,  z=2,  w  =  5,  ^=1. 
54.  We  pass  next  to  some  questions  producing  three  equations 
with  three  unknown  quantities. 

1.  Three  laborers  are  employed  in  a  certain  work.  A  and  B 
together  can  perform  it  in  8  days,'  A  and  C  together  in  9  days, 
and  B  and  C  together  in  10  days.  In  how  many  days  can  each 
alone  perform  the  same  work  ? 

Let  a;,  y  and  z  represent  the  number  of  days  respectively, 
then,  in  one  day  A  will  do  one-a:th  part  of  it,  B  one-yth  part, 
and  C  one-zth  part  of  it,  and  we  shall  have  for  the  equations  of 
the  question, 

z  ^  y         X  ^  z  y        z 


6P  ELEMENTS  OF   ALGEBRA. 

Dividing  the  first  of  these  equations  by  8,  the  second  by  9, 
and  the  third  by  10,  we  have 

1  ,  i__i  1    i_i  1  ,  i_2.. 

a;~^y"~"8'  x'^z^d'  y"^z""10' 
subtracting  the  second  equation  from  the  first,  and  adding  the 
third  to  the  remainder,  and  reducing,  we  obtain  y  ==  17|f ;  and 
in  like  manner  we  find  x  =  14f  |,  z  =  23/i-. 

2.  It  is  required  to  find  three  numbers,  such,  that  one-half  of 
the  first,  one-third  of  the  second,  and  one-fourth  of  the  third  shall 
together  make  46 ;  one-third  of  the  first,  one-fourth  of  the  second 
and  one-fifth  of  the  third  shall  together  make  35 ;  and  one-fourth 
of  the  first,  one-fifth  of  the  second  and  one-sixth  of  the  third, 
shall  together  make  28^. 

Ans.  12,  60,  and  80. 

3.  Three  brothers  purchased  an  estate  for  $15,000,  and  the 
first  wanted,  in  order  to  complete  his  part  of  the  payment,  half 
of  the  property  of  the  second ;  the  second  would  have  paid  his 
share  with  the  help  of  a  third  of  what  the  first  owned ;  and  the 
third  required,  to  make  the  same  payment,  in  addition  to  what 
he  had,  a  fourth  part  of  what  the  first  possessed ;  what  was  the 
amount  of  each  one's  property  ? 

Ans.  S3,000,  $4,000,  and  $4,250,  respectively., 

4.  Three  persons,  A,  B,  and  C,  compare  their  fortunes,  A 
says  to  B,  give  me  $700  of  your  money,  and  I  shall  have  twice 
as  much  as  you  retain ;  B  says  to  C,  give  me  $1400,  and  I  shall 
have  thrice  as  much  as  you  have  remaining ;  C  says  to  A,  give 
me  $420,  and  then  I  shall  have  5  times  as  much  as  you  retain. 
How  much  has  each  ? 

Ans.    A  $980,  B  $1540,  C  $2380. 

5.  Three  men,  A,  B,  C,  driving  their  sheep  to  market,  says  A 
to  B  and  C,  if  each  of  you  will  give  me  5  of  your  sheep,  I  shall 
have  just  half  as  many  as  both  of  you  will  have  left.  Says  B 
to  A  and  C,  if  each  of  you  will  give  me  5  of  yours  I  shall  have 
just  as  many  as  both  of  you  will  have  left.     Says  C  to  A  and 


EQUATIONS   OF   THE   FIRST   DEGREE.  ^ 

B,  if  each  of  you  will  give  me  5  of  yours  I  shall  have  just  twice 
as  many  as  both  of  you  will  have  left.    How  many  had  each  ? 

Ans.  10,  20,  and  30  respectively. 

6.  A  cistern  is  furnished  with  three  pipes,  A,  B  and  C.  By 
the  pipes  A  and  B  it  can  be  filled  in  12  minutes,  by  the  pipes  B 
^nd  C  in  20  minutes,  and  by  A  and  C  in  15  minutes.  In  what 
time  will  each  fill  the  cistern  alone,  and  in  what  time  will  it  be 
filled  if  all  three  are  open  together  ? 

Ans.  A  will  fill  it  in  20,  B  in  30,  and  C  in  60  minutes, 
and  the  three  together  in  10  minutes. 

7.  It  is  required  to  divide  the  number  72  into  four  such  parts, 
that  if  the  first  part  be  increased  by  5,  the  second  part  diminished 
by  5,  the  third  part  multiplied  by  5,  and  the  fourth  part  divided 
by  5,  the  sum,  difference,  product  and  quotient  shall  all  be 
e(^ual.  Ans.  The  parts  are  5,  15,. 2  and  50. 


SECTION  VI.— Negative  Quantities.     Questions  producing 
Negative  Results. 

55.  The  length  of  a  certain  field  is  eight  rods  and  the  breadth 
five  rods,  how  much  must  be  added  to  the  length,  that  the  field 
may  contain  30  square  rods  ? 

Let  z  =  the  quantity  to  be  added,  then  by  the  question 
40  + 5a;  =30, 
and  5a;  =  30  — 40,  * 

or  dividing  by  5  a;  =;  6  —  8. 

In  this  result  8,  the  quantity  to  be  subtracted,  is  greater  than 
that,  from  which  it  is  required  to  be  taken ;  the  subtraction  there- 
fore cannot  be  performed.     We  may,  however,  decompose  8  into 


7P  ELEMENTS   OF   ALGEBRA. 

two  parts  6  and  2,  the  successive  subtraction  of  which  will  be 
equal  to  that  of  8,  and  we  shall  have  for  6  —  8  the  equivalent 
expression,  6  —  6  —  2,  which  is  reduced  to  0  —  2  or  more  simply 
—  2,  the  sign  —  being  retained  before  the  2  to  show  that  it 
remains  to  be  subtracted. 

A  monomial  with  the  sign  —  prefixed  is  called  a  Tiega- 
five  quantity,  thus,  —  2,  —  3  a,  —  5  a  3,  are  negative  quanti- 
ties. 

Monomials  with  the  sign  -f-  either  prefixed  or  understood 
are  called  positive  quantities.  Thus,  2,  3 a,  5 ah  are  positive 
quantities. 

Negative  quantities,  it  will  be  perceived,  differ  in  nothing 
from  positive  quantities  except  in  their  sign.  They  are  derived 
from  endeavoring  to  subtract  a  larger  quantity  from  one  that  is 
smaller,  and  are  to  be  regarded  merely  as  positive  quantities  to 
be  subtracted. 

56.  If  it  now  be  asked  what  is  the  sum  of  the  monomials 
-j-  a,  —  3,  -|-  c,  the  question,  from  what  has  been  said,  is  reduced 
to  this,  what  change  will  be  produced  in  the  quantity  a,  if  the 
quantity  b  be  subtracted  from  it  and  the  quantity  c  be  added  to 
the  remainder.  Indicating  the  operations  required  to  obtain  the 
answer  to  the  question  thus  proposed,  the  result  will  be 

a  —  b-\-c. 
In  order  then  to  add  monomials  aflfected  with  the  signs  -f" 
and  — ,  it  will  be  sufficient  to  write  them  one  after  the  other 
with  the  signs  with  which  they  are  affected^  when  they  stand 
alon£. 

57.  If  we  now  add  the  quantities  -(-  i,  —  by  the  result  b  — bf 
it  is  evident,  will  be  equal  to  zero.  If  then  the  expression  b  —  b 
be  added  to  a,  it  will  not  affect  the  value  of  a ;  and  «--[-&  —  b 
will  only  be  a  different  form  of  expression  for  the  same  quantity 
a.  If  it  now  be  proposed  to  subtract  -|-  b  from  a,  it  will  be 
sufficient,  it  is  evident,  to  efface  -}-  ^  in  the  equivalent  expression 
a-j-i  —  b,  and  the   result  will  be  a  —  b.     Ag-ain,  if   it  be 


EQUATIONS    OF   THE    FIRST   DEGREE.  71 

required  to  subtract  — *  h  from  a,  it  will  be  sufficient  to  efface  —  b 
m  the  same  expression,  and  we  shall  have  for  the  result  a-\-h. 
Thus,  to  subtract  a  positive  quantity  is  the  same  as  to  add  an 
equal  negative  quantity,  and  to  subtract  a  negative  quantity  is 
the  same  as  to  add  an  equal  positive  quantity.  To  subtract  mo- 
nomials therefore  of  whatever  sign,  we  change  the  signs^  and 
then  proceed  as  in  addition. 

58.  If  we  multiply  b  —  bhj  a,  the  product  must  he  ab  —  ab^ 
because  the  multiplicand  being  equal  to  zero,  the  product  must 
be  zero.  Since  then  the  product  of  3  by  a  is  evidently  ab^ 
that  of  — bhj  a  must  be  —  a 3,  in  order  that  the  second  term 
may  destroy  the  first.  For  a  similar  reason  the  product  of  a  by 
b  —  b  will  be  ab  —  ab.  Whence  if  a  negative  quantity  be  mul' 
tiplied  by  a  positive,  or  a  positive  by  a  negative,  the  product  wilt 
be  negative. 

Again,  if  we  multiply  —  a  by  3  —  b,  from  what  has  been 
proved  above,  the  product  of  — ahj  b  will  be  —  ab,  the  product 
of  —  3  by  —  a  must  therefore  be  -\-  ab,  in  order  that  the  result 
may  be  zero,  as  it  should  be,  when  the  multiplier  is  zero. 
Whence,  the  product  of  a  negative  quantity  by  a  negative 
quantity  will  be  positive. 

The  rules  for  division  follow  necessarily  from  those  for  multi- 
plication. We  have  therefore  the  same  rules  for  the  signs  in 
the  multiplication  and  division  of  isolated  simple  quantities,  as 
are  applied  to  these  quantities,  when  they  make  a  part  of  poly- 
nomials ;  and  in  general,  monomials,  when  they  are  isolated  are 
combined  in  the  same  manner  with  respect  to  their  signs,  as  when 
they  make  a  part  of  polynomials. 

59.  From  what  has  been  said,  it  will  be  perceived,  that  the 
term  addition  does  not  in  algebra,  as  in  arithmetic,  always  imply 
augmentation.  Thus,  the  sum  of  a  and  —  b  is,  strictly  speaking, 
the  difference  between  a  and  b  ;  it  will  therefore  be  less  than  a. 
To  distinguish  this  from  an  arithmetical  sum,  we  call  it  an 
algebraic  sum.     Thus  the  polynomial 

Sab  —  5bc.-\-cd  —  ef, 


72  ELEMENTS    OF    ALGEBRA. 

considered  as  formed  by  uniting  the  quantities 
Sab,  —  53c, -|-  cd,  —  ef 
with  their  respective  signs,  is  called  an  algebraic  sum.  lu 
proper  acceptation  is  the  arithmetical  difference  between  the 
sum  of  the  units  contained  in  the  terms,  which  are  additive, 
and  the  sum  of  those  contained  in  the  terms,  which  are  sub- 
tractive. 

In  like  manner  the  term  subtraction  in  algebra  does  not 
always  imply  diminution.  Thus  —  b  subtracted  from  a  gives 
a-\-b,  which  is  greater  than  a.  This  result  may,  however,  be 
called  an  algebraic  difference,  since  it  may  be  put  under  the  form 
a-{-b). 

60.  Resuming  now  the  question  proposed,  art.  55,  we  havQ 
for  the  answer  x  =  —  2.  In  order  to  interpret  this  negativf 
result,  we  return  to  the  equation  of  the  question  40  -|-  52;  =  30 
Here,  the  addition  intended  in  the  enunciation  of  the  questioij 
being  arithmetical,  it  is  evidently  absurd  to  require  that  some 
thing  should  be  added  to  40  in  order  to  make  30,  since  40  iar 
already  greater  than  30.  The  negative  result  indicates,  there- 
fore, that  the  question  is  arithmetically  impossible,  or  in  other 
words,  that  it  cannot  be  solved  in  the  exact  sense  of  the  enuncia- 
tion. If,  however,  in  the  equation  40 -f- 5  a:  =  30,  we  substitute 
—  2  for  z,  we  have  40  — 10  =  30,  an  equation  which  is  exact. 
In  order  then  that  the  result  may  be  positive,  or  which  is  the 
same  thing,  that  the  question  may  be  arithmetically  possible,  the 
enunciation  should  be  modified  thus. 

The  length  of  a  certain  field  is  eight  rods,  and  its  breadth  five 
rods ;  how  much  must  be  subtracted  from  the  length,  that  the 
field  may  contain  30  square  rods  ? 

Putting  X  for  the  quantity  to  be  subtracted,  we  have  by  this 
new  enunciation  40  —  5  2:  =  30,  from  which  we  obtain  a:  =  2. 

2.  The  length  of  a  certain  field  is  11  rods  and  its  breadth  7 
rods ;  how  much  must  be  subtracted  from  the  length,  that  the 
field  may  contain  98  square  rods  ? 


EQUATIONS   OF  THE    FIRST   DEGREE.  tJI 

Let  a;=  the  quantity  to  be  subtracted ;  then  by  the  question 
77  — 7a:  =  98; 
whence  a:  =  —  3. 

To  interpret  this  resuh,  we  return  to  the  equation  of  the 
question.  Here,  as  an  arithmetical  subtraction  is  intended  in 
the  enunciation,  it  is  evidently  absurd  to  require,  that  some- 
thing should  be  subtracted  from  77  to  make  98,  since  77  is 
already  less  than  98.  The  question  therefore  cannot  be  solved 
in  the  exact  sense  of  the  enunciation.  If,  however,  instead 
of  X  in  the  equation  of  the  question,  we  substitute  —  3,  we 
have  77 -|- 21  =  98,  an  equation  which  is  exact.  In  order 
then  that  the  result  may  be  positive,  the  question  should  be 
modified,  thus, 

The  length  of  a  certain  field  is  11  rods  and  the  breadth  7  rods; 
how  much  must  be  aMedi  to  the  length  in  order  that  the  field 
may  contain  98  square  rods  ? 

Resolving  the  question  according  to  this  new  enunciation,  we 
obtain  a;  =  3. 

Let  us  take  as  a  third  example  the  following  question. 

3.  A  laborer  wrought  for  a  person  12  days  and  had  his  wife 
and  son  with  him  7  days,  and  received  46  shillings.  He  after- 
wards wrought  8  days,  having  his  wife  and  son  with  him  5  days, 
and  received  30  shillings ;  how  much  did  he  earn  per  day  him- 
self, and  how  much  did  his  wife  and  son  earn  ? 

Let  X  a=  the  daily  wages  of  the  man,  and  y  that  of  his  wife 
and  son ;  we  have  by  the  question 

12a; +  72/ =  46 
8a;-J-oy  =  30. 

Resolving  these  equations,  we  obtain  a;  =  5,  y  =  —  2. 

In  order  to  interpret  this  negative  result,  we  substitute  5  for  x 
in  the  equations  above,  by  which  we  have 
60 +  7?/==  46 
40  +  5^  =  30, 

G 


74  ELEMENTS    OF    ALGEBRA. 

equations  which  are  evidently  absurd,  since  it  is  required  to 
add  something  to  60  in  order  to  make  46,  and  to  40  in  order  to 
make  30.  If,  however,  we  substitute  —  2  for  y  in  these  last  we 
have 

60—14  =  46 
40  —  10  =  30, 
equations  which  are  exact.  The  negative  value  therefore  obtained 
for  y,  shows  that  the  allowance  made  for  the  wife  and  son  instead 
of  augmenting  the  pay  of  the  laborer,  should  be  regarded  as  a 
charge  placed  to  his  account.  The  question  therefore  should  be 
modified,  thus, 

A  laborer  wrought  for  a  person  12  days  and  had  his  wife  and 
son  with  Jiim  7  days  at  a  certain  expense^  and  received  46  shil- 
lings. He  afterwards  wrought  8  days,  having  his  wife  and  son 
with  him  5  days  at  expense  as  before,  and  received  30  shillings. 
How  much  did  the  laborer  earn  per  day,  and  how  much  was 
charged  him  per  day  on  account  of  his  wife  and  son  ? 

Resolving  the  question,  thus  stated,  we  have 
a:=5,2/  =  2. 

61.  From  what  has  been  done,  it  will  be  perceived,  that  in 
problems  of  the  first  degree,  a  negative  result  indicates  some 
inconsistency  in  the  enunciation  of  the  question,  arithmetically 
considered,  and  at  the  same  time  shows  how  this  inconsistency 
may  be  reconciled  by  rendering  subtractive  certain  quantities, 
which  had  been  regarded  as  additive,  or  additive  certain  quanti- 
ties, which  had  been  regarded  as  subtractive. 

Negative  results,  however,  in  the  extended  sense,  in  which 
the  terms  addition  and  subtraction  are  used  in  algebra,  may 
be  regarded  as  answers  to  questions.  Thus,  in  the  equation 
40  -[-  5a:  ==  30,  the  negative  result  — 2  shows  that  it  is  necessa- 
ry to  add  — 10  to  40  to  obtain  30.  By  means  of  this  exten- 
sion of  the  meaning  of  the  terms,  addition  and  subtraction, 
we  may  regard  as  one  single  question,  those,  the  enunciations 
of  which   are  such,  that  the   solution,  which  satisfies  one  of 


EQUATIONS    OF   THE    FIRST    DEGREE.  75 

the  enunciations,  will  by  a  mere  change  of  sign  satisfy  the  other 
also. 

62.  The  following  examples  will  serve  as  an  exercise  in  the 
interpretation  of  negative  resuUs. 

1.  A  father  is  55  years  old,  and  his  son  is  16.  In  how  many 
years  will  the  son  be  one-fourth  as  old  as  the  father  ? 

2.  What  number  is  that,  whose  fourth  part  exceeds  its  third 
part  by  12  ? 

3.  There  are  two  numbers  such,  that  if  twice  the  second  be 
added  to  the  first,  the  sum  will  be  20,  but  if  3  times  the  second 
be  subtracted  from  the  first,  the  difference  will  be  45.  What  are 
the  numbers  ? 

4.  To  divide  the  number  40  into  two  such  parts,  that  if  the 
first  be  multiplied  by  7  and  the  second  by  5,  the  sum  of  the 
products  will  be  90. 

5.  What  number  must  be  subtracted  from  the  numbers  70 
and  50  respectively  in  order  that  their  differences  may  be  as  4 
to  3? 

6.  Three  persons  comparing  their  property,  it  is  found,  that 
A's  and  B's  together  amount  to  $1000,  A's  and  C's  to  $480,  and 
B's  and  C's  to  $400.     What  amount  of  property  has  each  ? 

7.  A  laborer  wrought  for  a  gentleman  10  days,  having  his 
wife  with  him  5  days  and  his  son  4  days,  and  received  $14,25. 
At  another  time  he  wrought  8  days,  having  his  wife  with  him  6 
days  and  his  son  3  days,  and  received  $13.  At  a  third  time  he 
wrought  6  days,  having  his  wife  with  him  4  days  and  his  son  5 
days,  and  received  $8,00.  How  much  did  he  receive  a  day 
himself,  and  how  much  for  his  wife  and  son  severally  ? 


76  ELEMENTS    OF   ALGEBRA. 


SECTION  VII.— Indeterminate  Analysis. 

63.  Let  it  be  proposed  to  find  two  numbers  such,  that  the  first 
added  to  three  times  the  second  shall  be  equal  to  15. 

Putting  z  and  y  for  the  numbers  sought,  we  have  by  the 
question 

a:  +  32/=15; 
here  as  we  have  two  unknown  quantities  and  but  one  equation, 
the  particular  numbers  intended  in  the  question  proposed  cannot 
be  determined.     Deducing,  however,  from  the  equation  the  value 
of  one  of  the  unknown  quantities,  x  for  example,  we  have 

a:=15  — Sy. 
If  we  now  assume  arbitrarily  any  values  whatever  for  y,  we  shall 
obtain  values  for  a:,  which  will  satisfy  the  equation, 

thus,  let  2/==       1,       IJ,      2,      SJ 

webave  a:=     12,     lOJ,      9,      8 

or  otherwise      t/  =  —  1,  —  1  J,  —  2,  —  2| 

wehave  a;=    18,    19J,    21,     22 

pairs  of  values  for  x  and  y^  which,  it  is  easy  to  see,  will  satisfy 
the  equation,  and  the  number  of  w^hich  may  be  increased  without 
limit. 

In  general,  if  the  conditions  of  a  problem  furnish  fewer 
equations,  than  there  are  unknown  quantities  to  be  determined, 
the  equations  of  the  problem  will  admit  of  an  infinity  of  systems 
of  values  for  the  unknown  quantities,  if  we  understand  by  these 
any  quantities  whatever,  entire  or  fractional,  positive  or  negative. 
It  is  frequently  the  case,  however,  that  the  nature  of  the  question 
requires,  that  the  values  of  the  unknown  quantities  should  be 
entire  numbers.  This  circumstance,  it  is  evident,  will  very 
much  restrict  the  number  of  solutions,  especially  if  we  reckon 
the  direct  solutions  only,  that  is  to  say,  solutions  in  entire  and 
positive  numbers. 

Thus,  if  in  the  question  proposed  the  numbers   sought  are 


INDETERMINATE    ANALYSIS.  77 

required  to  be  entire  and  positive,  the  value  of  y,  it  is  evident, 
must  not  exceed  5 ;  if  then  we  put  successively  for  y 

2^  =  0,     1,2,3,4,5 
we  have  x  =  15, 12,  9,  6,  3,  0, 

and  the  question  admits  of  six  different  solutions  only,  the 
solution  in  which  0  is  reckoned  as  a  value  of  one  of  the  unknown 
quantities  being  included. 

Problems  of  the  kind,  which  we  are  here  considering,  are 
called  indeterminate  problems^  and  that  part  of  algebra,  which 
relates  to  the  solution  of  indeterminate  problems,  is  called  inde- 
terminate analysis. 

64.  The  preceding  question,  in  which  the  coefficient  of  one 
of  the  unknown  quantities  is  equal  to  unity,  presents  no  diffi- 
culty. We  shall  now  show,  that  whatever  the  coefficients  of  the 
unknown  quantities,  the  solution  of  the  question  proposed  may 
be  made  to  depend  upon  the  solution  of  an  equation,  in  which 
the  coefficient  of  one  of  the  unknown  quantities  is  equal  to 
unity. 

Let  it  be  proposed  then  to  find  the  entire  values  of  x  and  y  m 
the  equation  17 a:  =  542  —  \\y. 

Deducing  from  this  equation  the  value  of  y,  we  have 

542— 17  a: 

or  performing  the  division  as  far  as  possible,  we  have 

._  ,    3  — 6a; 

y==49~a:+-yp-. 

But,  by  the  question  the  values  of  x  and  y  should  be  entire 

numbers,  it  is  necessary^  therefore,  and  it  is  sufficient ^  that  — rrr— 

should  be  equal  to  an  entire  number.    Let  t  be  this  number 
{t  is  called  an  indeterminate,)  we  have 
y  =  49  — a;  +  ? 
1U=  3  — 6ar.        (2) 


78  ELEMENTS    OF   ALGEBRA. 

and  the  question  is  now  reduced  to  resolve  in  entire  numbers 
the  equation  (2),  the  coefficients  of  which  are  more  simple 
than  those  of  the  proposed.  Deducing  from  this  equation  the 
value  of  X  and  performing  the  division  as  far  as  possible,  we 
have 

Here,  since  x  and  t  are  entire  numbers   — - —  must  be  equal 

to  an  entire  number ;  let  t'  be  this  number,  the  letter  t  being 
marked  with  an  accent  to  show  that  it  represents  a  quantity 
different  from  that  before  represented  by  it,  we  have 

6f  =  3  —  5t,        (3) 
And  the  question  is  still  further  reduced  to  resolve  in  entire 
numbers  the  equation  (3),  the  coefficients  of  which  are  more 
simple  than  those  of  equation  (2).     Deducing  from  this  equation 
the  value  of  t,  we  have 

2 f 

but  t  and  f  in  this  equation  are  entire  numbers ;  — ^ —  must 

therefore  be  equal  to  an  entire  number ;  let  f  be  this  number, 
we  have  t  =  —t.'-^t" 

or  f  =  S  —  5t",        (4) 

and  the  question  is  now  reduced  to  resolve  in  entire  numbers 
the  equation  (4),  in  which  the  coefficient  of  one  of  the  unknown 
quantities  t'  is  equal  to  unity.  Indeed,  the  two  principal  unknown 
quantities  and  the  several  indeterminates  employed  are,  it  is 
evident,  connected  together  by  the  equations 
y  =  49  —  a;  +  ^ 

X=:'—t']-f 


INDETERMINATE    ANALYSIS.  79 

if  then  we  give  any  entire  value  whatever  to  t"  and  return  to 
the  values  of  x  and  y  corresponding,  the  values  thus  found,  it  is 
evident,  will  be  entire  numbers,  and  will  satisfy  the  equation 
proposed.  Thus,  let  ^"  =  1,  we  have  a:  =  —  5, 2/  =  57,  values 
which,  it  is  easy  to  see,  will  satisfy  the  equation  proposed. 

To  determine  with  more  facility  the  values  of  t"  which  will 
give  entire  values  for  x  and  y,  we  express  x  and  y  immediately 
in  teirms  of  t".  In  order  to  this  we  substitute  for  t'  its  value  in 
the  equation  for  ?,  which  gives 

?  =  — (3  — 50  +  ^"  =  6r  — 3. 

Substituting  next  for  t  and  f  their  values  in  the  equation  for  a:, 
and  for  x  and  t  their  values  in  the  equation  for  y,  we  obtain 
finally  a;  =  6— llif" 

2^  =  40-(-17r. 
If  then  we  make  successively  ^"=0,  1,  2,  3,  .  .  .  or  other- 
wise, ?"=0, —  1,  —  2,  —  3,  ...  in  the  above,  we  shall  ob- 
tain all  the  entire  values  of  x  and  y  proper  to  satisfy  the  equa- 
tion proposed.  But  if  entire  and  positive  values  only  are  re- 
quired for  X  and  y,  it  will  be  necessary  to  give  to  t"  such  values 
only,  as  will  render  6 — 11  if",  40 -j- 17  if"  positive.  It  is  evi- 
dent, that  ^"  =  0,  t"  =  —  1,  ^"  =  —  2,  are  the  only  values  of 
t",  that  will  fulfil  this  condition ;  for,  every  positive  value  of 
t"  will  render  x  negative,  and  every  negative  value  of  t"  nu- 
merically greater  than  2,  willvrender  y  negative.  Putting  there- 
fore if"=  0,  —  1,  —  2  successively,  we  have 

a;  =  6,    17,28 
2,  =  40,  23,    6. 

The  proposed  therefore  admits  of  three  different  solutions  in 
entire  and  positive  numbers,  and  of  three  only. 

2.  Let  it  be  proposed,  as  a  second  example,  to  divide  the 
number  159  into  two  such  parts,  that  the  first  may  be  divisible 
by  8,  and  the  second  by  13. 

Designating  by  x  and  y  the  quotients,  arising  from  the  division 


80  ELEMENTS    OF    ALGEBRA. 

of  the  parts  sought  by  the  numbers  8  and  13  respectively,  we 
have  by  the  question  8a; -f-  137/=  159. 

Pursuing  with  this  equation  the  same  process,  as  in  the  pre- 
ceding example,  we  have  the  five  equations 

a;  =  19  — 2/  +  ^ 
y=    l  —  t^-f 

Expressing  x  and  y  in  terms  of  t"\  we  have 

a:=15+13r' 

y=   3  — Sr'. 

Here  it  is  evident,  that  t'"  =  0,  and  t'"  =  —  1  are  the  only 
values  of  t"\  which  will  give  entire  and  positive  values  for  z 
and  y.     Making  successively  it"'=  0,  t'"  =  —  1,  we  have 

a:=15,2 
2^  =  3,11. 

Since  then  82;  and  \2y  represent  the  parts  required,  the 
proposed  admits  of  two  solutions,  viz.  120  and  39  for  the  first 
solution,  and  16  and  143  for  the  second. 

The  expressions,  a:=  15+ 13^',  2/  =  3  — 8^',  are  called 
formulas  for  x  and  ?/,  since  they  indicate  the  manner  in  which 
the  values  of  x  and  y  are  obtained.  In  the  use  of  these  formulas 
the  accents,  it  is  evident,  may  be  omitted,  as  we  have  now  no 
further  occasion  to  distinguish,  one  from  the  other,  the  indeter- 
minates  which  have  been  employed. 

3.  It  is  required  to  divide  25  into  two  parts,  one  of  which  may 
be  divisible  by  2,  and  the  other  by  3. 

Ans.  The  parts  are  16  and  9,  10  and  15,  or  4  and  21. 

4.  A  person  has  in  his  pocket  pieces  of  5  shillings  and  3 
shillings  only,  and  wishes  to  pay  a  bill  of  58  shillings.  How 
many  pieces  must  he  give  of  each  ? 

Ans.  2  of  the  first  and  16  of  the  secondj  or  &c« 


INDETERMINATE   ANALYSIS.  81 

5.  A  sum  of  $81  was  distributed  among  some  poor  persons, 
men  and  women;  each  woman  received  $5,  and  each  man  S7. 
How  many  men  and  women  were  there  ? 

Ans.  There  were  3  men  and  12  women,  or  &c. 

65.  Let  it  be  required  next  to  solve  in  entire  numbers  the 
equation  49a;  —  S5y  =11. 

Here  it  will  be  observed,  that  the  coefficients  of  x  and  y 
have  a  common  factor  7;  dividing  therefore  both  members  by 

7,  we  have  7x  —  6y=.  —  ^  an  equation  which  is  evidently  im- 
possible in  entire  numbers ;  the  proposed  therefore  does  not 
admit  of  entire  and  positive  values  for  x  and  y.  In  general, 
the  proposed  equation  being  reduced  to  the  form  of  the  pre- 
ceding, if  the  coefficients  of  x  and  y  have  a  common  factor ^  which 
does  not  enter  into  the  second  member ^  the  equation  is  impossible 
in  entire  numbers. 

If  there  be  a  factor,  common  to  the  coefficients  of  x  and  y, 
which  does  not  enter  into  the  second  member,  and  this  factor 
be  not  perceived  at  first,  the  course  of  the  calculation  will  make 
known,  sooner  or  later,  the  impossibility  of  solving  the  question 
in  entire  numbers. 

Applying  the  process,  explained  above,  to  the  equations 
49 X  —  35?/ =11,  we  obtain  finally  the  equation 

an  equation,  which  is  evidently  impossible  in  entire  numbers 
for  t  and  t\  from  which  it  is  readily  inferred,  that  the  proposed 
.  will  not  admit  of  entire  solutions. 

If  the  equation  of  the  proposed  question  has  therefore  a  factor 
common  to  both  members,  we  suppress  it;  the  coefficients  of 
z  and  y  will  then  be  prime  to  each  other,  if  the  question  admits 
of  solution  in  entire  and  positive  numbers.  This  being  the  case, 
the  process  explained  above  will  always  lead  to  a  final  equation, 
in  which  the  coefficient  of  one  of  the  indeterminates  is  equal  to 
6 


82  ELEMENTS    OF    ALGEBRA. 

unity.  Indeed,  it  will  readily  be  perceived,  that  in  the  course 
pursued  we  apply  to  the  coefficients  of  x  and  y  in  the  proposed 
the  process  of  the  greatest  common  divisor;  since  then  these 
coefficients  are  by  hypothesis  prime  to  each  other,  we  arrive 
finally  at  a  remainder  equal  to  unity,  which  will  be  the  coefficient 
of  the  last  but  one  of  the  indeterminates  introduced  in  the  course 
of  the  calculation. 

66.  In  certain  cases  the  preceding  process  admits  of  simpli- 
fications, which  it  is  important  to  introduce  in  practice. 

1.  Let  it  be  required  to  solve  in  entire  numbers  the  equa- 
tion 5a:  +  3?/ =  49. 

Proceeding  as  before,  we  have 

32/  =  49  — 5a; 
y=^lQ^x 3—;^ 

but  the  quotient  on  dividing  5  a;  by  3  being  nearer  2a;,  we  put 
the  equation  under  the  form 

3y  =  49  — 6a;  +  a;; 

x-\- 1 
whence  y=16  —  2x-\ ?— J 

from  which  we  obtain      a;  =  3  ^  —  1 
y=18  — 5^. 

By  means  of  the  simplification,  here  introduced,  the  number 
of  indeterminates,  employed  in  the  calculation,  is  one  less,  than 
would  otherwise  be  necessary. 

2.  A  person  purchases  wheat  at  I65.  and  barley  at  9*.  a 
bushel,  and  pays  in  all  167s.  How  many  bushels  of  each  did 
he  purchase  ?  Ans.  2  of  wheat  and  15  of  barley. 

3.  To  find  two  numbers  such,  that  if  the  first  be  multiplied 
by  7  and  the  second  by  13,  the  sum  of  the  products  will  be 
128.  Ans.  The  numbers  are  9  and  5,  or  &c. 

4.  Let  it  be  proposed  next  to  resolve  in  entire  numbers  the 
equation  13  a;  -—  57y  =  101. 


INDETERMINATE   ANALYSIS.  83 

Deducing  the  value  of  z,  we  have 

a:  =  4y  +  7+-^^^  =  4y  +  7+       ^J    ^ 
In  order  that  a;  and  y  in  this  equation  may  be  entire  num- 
bers,    ^7^      must  be  equal  to  an  entire  number;  but  since 
5  and  13  are  prime  to  each  other,  it  is  necessary  in  order  to 

y     j     o 

this,  that  •^  Jl     should  be  an  entire  number ;  putting  t  for  this 

number,  we  have 

a;  =  4y-|-7  +  5^ 
13^=    y  +  2. 
from  which  we  obtain 

a:  =  57^— 1 

Here,  every  entire  and  positive  value  for  t  will  give  similar 
values  for  x  and  y  ;  but  if  we  suppose  ^  =  0,  or  to  be  negative, 
the  values  of  x  and  y  will  be  negative.  The  number  of  entire 
and  positive  solutions  of  the  proposed  is  therefore  infinite^  and 
the  smallest  system  of  values  for  x  and  y  is 
a;  =  56, 2/=ll. 

By  means  of  the  simplifications,  here  introduced,  one  inde- 
terminate  only  is  employed,  instead  of  three,  which  would  other- 
wise be  necessary. 

5.  To  divide  100  into  two  such  parts,  that  if  the  first  be 
divided  by  5,  the  remainder  will  be  2;  and  if  the  second  be 
divided  by  7  the  remainder  will  be  4. 

Ans.  The  parts  are  47  and  53,  or  12  and  88. 

6.  To  find  two  numbers  such,  that  11  times  the  fitst  dimin- 
ished by  7  times  the  second,  may  be  equal  to  53. 

Ans.  8  and  5,  or  &c. 

7.  A  person  purchases  some  horses  and  oxen;  he  pays  $30 
for  each  horse,  and  $23  for  each  ox;  and  he  finds,  that  the  oxen 
cost  him  $12  more  than  the  horses.  How  many  horses  and 
oxen  did  he  buy  ?  Ans.  18  horses  and  24  oxen,  or  &c. 


Bi  ELEMENTS   OF  ALGEBRA. 

8.  To  find  two  numbers  such,  that  if  8  be  added  to  17  tirKies 
the  first,  the  sum  will  be  equal  to  49  times  the  second. 

Ans.  37  and  13,  or^&c. 
#,  Let  it  be  proposed  next  to  resolve  the  equation 

39  a:  — 56  2/ =11. 
JDeducing  from  this  equation  the  value  of  x,  we  have 

^^y^    39    * 

Here,  yi  the  expression  — ~J^ — ,  it  will  be  observed  that  the 

difference  between  17,  the  coefficient  of  y,  and  the  divisor  39, 
contains  the  other  term  11  as  a  factor;  on  this  account  we  take 
the  quotient  56 y  divided  by  39  in  excess,  which  gives 
^_g^      22y-ll_  ll(2y-l). 

^  —  '^y 39^  """^^  39         ' 

from  which  we  readily  obtain 

a:  =  56?'  — 27 
y  =  39j^'— 19. 

10.  To  find  two  numbers  such,  that  if  the  first  be  multiplied 
by  11  and  the  second  by  17,  the  first  product  is  5  greater  than 
the  second.  Ans:  19  and  12,  or  &c. 

11.  In  how  many  ways  can  a  debt  of  546  livres  be  paid,  by 
paying  pieces  of  15  livres,  and  receiving  in  exchange  pieces  of 
11  livres?        Ans.  The  number  of  ways  is  infinite.     For  the 

first  we  have  43  of  the  one,  and  9  of  the  other. 

12.  The  difference  between  two  numbers  is  309,  and  if  the 
greater  be  divided  by  37  the  remainder  will  be  5,  and  if  the 
less  be  divided  by  54  the  remainder  will  be  2;  what  are  the 
numbers  ?  Ans.  1337  and  1028,  or  &c. 

67.  From  what  has  been  done,  it  will  be  perceived,  that  if  the 
equation  proposed  be  of  the  form  2a;-f-3y  =  10,  the  number 
of  solutions  in  entire  and  positive  numbers  will  be  limited;  but 
if  the  equation  be  of  the  form  2  a:  — 3y=10,  10  being  either 
positire  or  negative,  the  number  of  solutions  will  be  infinite. 


INDETERMINATE   ANALYSIS.  05 

If  moreover  we  compare  the  formulas  for  x  and  y  with  the 
equations  from  which  they  are  derived,  the  coefficient  of  the 
indeterminate  in  the  formula  for  x  is  the  same,  it  will  be  ob- 
served, with  the  coefficient  of  y  in  the  equation;  and  the  co- 
efficient of  the  indeterminate  in  the  formula  foT  y  is  the  same 
with  the  coefficient  of  x  in  the  equation,  taken  with  the  con- 
trary sign,  or  the  converse,  as  it  respects  the  signs  of  the  co- 
efficients. Having  obtained  then  a  first  solution  of  the  ques- 
tion proposed,  those  which  follow  will  be  found  by  adding 
successively  to  the  values  of  x  the  coefficient  of  y  in  the  equa- 
tion, and  subtracting  successively  from  the  values  of  y  tlie  co- 
efficient of  X  in  the  equation,  or  the  converse,  the  coefficients 
of  X  and  y  being  taken  with  the  signs,  which  they  have  in  the 
equation. 

68.  We  pass  next  to  the  solution  of  problems  and  equations 
with  three  or  more  unknown  quantities. 

1.  Let  it  be  proposed  to  pay  741  livres  with  41  pieces  of 
money  of  three  different  species,  viz.  pieces  of  24  livres,  19 
livres,  and  10  livres. 

Let  X,  y  and  z  represent  respectively  the  number  of  pieces  of 
each  kind,  we  have  by  the  question 

a:  +  2/-|-^  =  41 
24a:  +  192/  +  10z  =  74L 

Eliminating  one  of  the  unknown  quantities,  x  for  example,  we 
have  5y+14z  =  243. 

Deducing  from  this  equation  formulas  for  entire  values  for  z 
and  y,  according  to  the  method  explained  above,  we  have, 
omitting  the  accents, 

2  =  5^  —  3 

y  =  57  —  UU 

Substituting  next,  in  the  first  of  the  equations  of  the  proposed 
the  expressions  for  z  and  y  just  obtained,  and  deducing  the 
▼alue  of  X,  we  have     x  =  9t  —  13. 


86  ELEMENTS   OF   ALGEBRA. 

If  we  now  put  for  f,  in  the  above  formulas  for  ar,  y,  and  «, 
any  entire  values  whatever,  we  shall  obtain  entire  values  for 
Xt  y,  and  z,  which  will  satisfy  the  equations  of  the  proposed. 
But  to  obtain  the  entire  and  positive  values  only,  as  the  nature 
of  the  question  requires,  it  is  evident,  1°.  that  9t  must  be 
greater  than  13,  or  which  is  the  same  thing,  that  t  must  be 
greater  than  IJ;  2**.  that  14^  must  be  less  than  57,  or  which 
is  the  same  thing,  that  t  must  be  less  than  4^ ;  t  can  therefore 
have  only  the  values  of  2,  3,  4. 

Putting  in  the  formulas  above  t  equal  to  2,  3,  and  4  succes- 
sively, we  have 

x=  5,14,23 
3^  =  29,15,  1 
z=   7,  12,17. 

The  proposed,  therefore,  admits  of  three  different  solutions 
and  of  three  only. 

2.  Thirty  persons,  men,  women,  and  children,  spend  80 
crowns  in  a  tavern.  The  share  of  a  man  is  7  crowns,  that  of 
a  woman  5  crowns,  and  that  of  a  child  is  2  crowns.  How 
many  persons  were  there  of  each  class  ? 

Ans.  For  a  first  answer  we  have,  1  man, 
5  women,  and  24  children. 

3.  The  sum  of  three  entire  numbers  is  15,  and  if  the  first 
be  multiplied  by  2,  the  second  by  3  and  the  third  by  7,  the  sum 
of  the  products  will  be  65.     What  are  the  numbers  ? 

Ans.  4,  5,  and  6. 

From  what  has  been  done,  it  will  be  easy  to  see  how  we  are 
to  proceed,  in  the  case  of  three  equations  with  four  unknown 
quantities,  and  so  on. 


SOLUTION   OF    QUESTIONS   IN   A   GENERAL   MANNER.  87 

SECTION  VIII. — Solution  of  Questions  in  a  General 
Manner. 

69.  In  the  solution  of  a  question  m  numbers,  there  are,  it 
must  have  been  perceived,  two  distinct  things  which  require 
attention.  1°.  To  determine  by  a  process  of  reasoning  what 
operations  must  be  performed  upon  the  numbers  given  in  the 
question  in  order  to  obtain  the  answer  sought;  2°.  to  perform 
these  operations.  In  the  questions  which  have  been  solved 
thus  far,  the  operations  have  each  been  performed  as  soon  as 
determined.  Let  us  now  resume  the  question,  art.  1,  and  in- 
stead of  performing  the  operations,  as  we  proceed,  let  us  retain 
them  by  means  of  the  proper  signs. 

Representing  as  before  the  less  part  by  x,  the  greater  will 
be  a;  -j-  12,  and  we  have 

x-\-x+l2  =  5Q 

2x=:56  —  l2 
56      12 
^=2—2- 
Here  the  process  of  reasoning  required  in  the  solution  of 
the  proposed  has  been  conducted  by  itself;  the  expression,  at 
which  we  arrive,  is  not  the  answer  sought,  but  the  result  of  the 
reasoning  pursued ;  it  shows  what  operations  must  be  performed 
in  order  to  obtain  the  answer,  viz.  that  from  one  half  of  56,  the 
number  given  to  be  divided,  there  must  be  subtracted  one  half 
of  12  the  given  excess.     Performing  next  the  operations   thus 
determined,  we  have  22  for  the  less  part  as  before. 

Let  us  next  resume  the  sixth  question,  art.  6;  representing 
again  the  least  part  by  x,  the  mean  will  he  x -\-  40,  the  great- 
est a;  -|-  40  -[-  60,  and  we  shall  have 

a:  +  z  +  40  +  a:  +  40  +  60  =  230 

8a:  =  230  — 40  — 40  — 60 

3a:  =  230—  2  x40  — 60 

230—2  X  40  —  60 


S3  ELEMENTS    OF   ALGEBRA. 

Here  also  the  result  shows  the  operations  to  be  performed, 
according  to  which  to  find  the  least  part  sought,  from  230,  the 
number  given  to  be  divided,  we  subtract  twice  40,  or  twice  the 
excess  of  the  mean  part  above  the  least,  and  also  60,  the  excess 
of  the  greatest  part  above  the  mean,  and  take  one  third  of  the 
remainder. 

70.  If  the  reasoning  pursued  in  the  solution  of  the  preceding 
questions  be  examined  with  attention,  it  will  be  perceived,  that 
it  does  not  depend  upon  the  particular  numbers  given  in  these 
questions.  It  will  be  precisely  the  same  for  any  other  numbers. 
The  same  operations  will  therefore  be  necessary  to  obtain  the 
parts  sought,  whatever  the  given  number  may  be. 

By  preserving  the  operations,  therefore,  we  resolve  the  pro- 
posed in  a  general  Tnanner^  that  is,  we  determine  once  for  all 
what  operations  are  necessary  for  all  questions,  which  differ 
from  the  proposed  only  in  the  particular  numbers,  which  are 
given. 

Let  it  be  proposed  next  to  find  a  number  such  that  the  differ- 
ence between  one-ninth  and  one-seventh  of  this  number  shall  be 
equal  to  10. 

Putting  X  for  the  number,  we  obtain 

9a;  — 7a;  =  7X  9X  10. 

Here  it  will  be  observed  that  x  is  taken  9  times  minus  seven 
times,  or  9  —  7  times ;  9  —  7  will,  therefore,  be  the  coefficient 
of  a;,  and  the  above  equation  may  be  written  thus : 

(9  — 7)  a:  =.7X9X10 

,  7  X  9  X  10 

whence  x  =  — 5- — = — . 

Let  the  following  questions  be  now  resolved  in  a  general 
manner. 

L  A  company  settling  their  reckoning  at  a  tavern,  pay  8s. 
each ;  but  if  there  had  been  4  persons  more,  they  should  only 
have  paid  7*.  each.     How  many  were  there  ? 


SOLUTION  OF   QUESTIONS  IN  A  GENERAL  MANNER.  89 

.   2.  Divide  the  number  91  into  two  such  parts,  that  6  times  the 
first,  diminished  by  5  times  the  second,  may  be  equal  to  40. 

3.  Divide  the  number  56  into  two  such  parts,  that  one  part 
being  divided  by  7  and  the  other  by  3,  the  quotients  may  together 
be  equal  to  10. 

71.  In  the  solution  of  questions  in  a  general  manner,  accord- 
ing to  the  method  above  explained,  we  should  be  liable,  through 
inadvertence,  to  perform  some  of  ihe  operations  as  we  proceed ; 
thus  the  result  would  not  show  how  the  answer  is  to  be 
found  by  means  of  the  numbers  originally  given  in  the  question. 
To  avoid  this  inconvenience  and  at  the  same  time  to  render  the 
solution  more  concise,  it  is  usual  to  represent  the  given  things 
in  a  question  by  signs,  which  will  stand  indifferently  for  the 
particular  numbers  given  in  the  question,  or  for  any  other 
numbers  whatever. 

It  is  agreed  to  represent  known  quantities,  or  those  which 
are  supposed  to  be  given  in  a  question,  by  the  first  letters  of  the 
alphabet,  as  a,  i,  c. 

The  given  things  in  the  question,  art.  1,  are  the  number  to  be 
divided,  and  the  excess  of  the  greater  part  aboVe  the  less ;  rep 
resenting  these  by  a  and  b  respectively,  the  question  may  be 
presented  generally,  thus ;  To  divide  a  number  a  into  two  stcch 
parts  that  the  greater  may  exceed  the  less  by  b. 

To  resolve  the  question,  thus  stated,  we  denote  still  the  less 
part  by  x  ;  the  greater  will  then  be  a;  -{-  ^>  and  we  have 
x-{-x-\-b  =  a 
2x-\'b  =  a 

2x  =  a  —  b 

a      b 

^=2-2 

The  translation  of  a  formula  into  common  language  is  called 
a  rule.  Thus  we  have  the  following  rule,  by  which  to  obtain 
the  less  of  the  parts  required  according  to  the  question  pro- 


90  ELEMENTS    OF   ALGEBRA. 

posed,  viz.     From  half  the  number  to  he  divided^  subtract  half 
the  given  excess,  the  remainder  will  be  the  answer. 

Knowing  the  less  part,  we  obtain  the  greater  by  adding  to  the 
less  the  given  excess.  We  may,  however,  easily  obtain  a  rule 
for  calculating  the  greater  part  without  the  aid  of  the  less. 
Indeed  since  the  less  part  is  equal  to 

-  —  -,  if  we  add  b  to  this,  we  have  ^  —  o~t~^  equal  to  the 

greater.     But  this  expression  may,  it  is  easy  to  see,  be  reduced 

to  Q  +  o  5  whence  we  have  the  following  rule,  by  which  to  find 

the  greater  part,  viz.     To  half  the  number  to  be  divided,  add 
half  the  given  excess,  the  result  will  be  the  answer. 

To  apply  these  rules,  let  it  be  required  to  divide  $1753 
between  two  men  in  such  a  manner,  that  the  first  may  have 
$325  more  than  the  second. 

72.  The  6th  question,  art.  6,  may  be  presented  in  a  general 
manner,  thus ;  To  divide  a  number  a  into  three  such  parts,  that 
the  excess  of  the  mean  above  the  least  may  be  b,  and  the  excess 
of  the  greatest  above  the  mean  may  be  c. 

Let  X  =  the  least  part ; 

then  x-\-b=-  the  mean, 

and  x-\-b-\-c  =  the  greatest, 

therefore        x-\-x-\-b-\-x-\-b-\-c  =  a, 

or  transposing  and  reducing      2x  =  a  —  2b  —  c, 

a  —  2b  —  c 
whence  x  == ^ . 

Translating  the  above  formula  into  common  language,  we  have 
the  following  rule,  by  which  to  find  the  least  part,  viz.  From  the 
number  to  be  divided,  subtract  twice  the  excess  of  the  mean  part 
above  the  least,  and  also  the  excess  of' the  greatest  above  the  mean 
and  take  a  third  of  the  remainder. 

To  obtain  a  formula  for  the  mean  part,  we  add  b,  the  excess 


SOLUTION   OF   QUESTIONS   IN   A   GENERAL    MANNBR.  91 

of  the  mean  above  the  least,  to  the  above  expression  for  the  leas» 

part,  which  gives  for  the  mean 

a  —  2b  —  c    ,    . 
3- +  *. 

or  reducing  to  a  common  denominator 
g  — 23  — c  ,  3^ 
3  +"3  ' 

whence  we  obtain  for  the  mean  part 
a-\'b  —  c 
3       * 

In  like  manner  the  following  formula  will  readily  be  obtained 
for  the  greatest  part,  viz. 

g-|-3  +  2c 
/^  3" 

Translating  these  formulas  into  common  language  we  obtain 
rules  also  for  the  mean  and  for  the  greatest  part. 

1.  To  apply  these  rules  let  it  be  required  to  divide  $973  among 
three  men,  so  that  the  second  shall  have  $69  more  than  the  first, 
and  the  third  $43  more  than  the  second. 

2!  A  father,  who  has  three  sons,  leaves  them  his  property 
amounting  to  $15730.  The  will  specifies,  that  the  second  shall 
have  $2320  more  than  the  third,  and  the  eldest  shall  have  $3575 
more  than  the  second.     What  is  the  share  of  each  ? 

73.  The  operations  necessary  for  the  solution  of  this  last  ques- 
tion are,  it  is  easy  to  see,  the  same  with  those  for  the  preceding. 
It  may  therefore  be  solved  by  the  same  formulas.  In  like  man- 
ner the  seventh  and  eighth  questions,  art.  6,  may  be  solved  by 
the  same  formulas.  This  circumstance  is  worthy  attention,  since 
we  are  thus  enabled  to  comprehend  in  one  the  solution  of  a  multi- 
tude of  questions,  differing  from  each  other  not  only  in  the  par- 
ticular numbers,  which  are  given,  but  also  in  the  language,  in 
which  they  are  expressed. 

Let  now  the  following  questions  be  generalized. 

1.  The  sum  of  $3753  is  to  be  divided  among  4  men,  in  such 
a  manner,  that  the  second  will  have  $159  more  than  the  first,  the 


92  ELEMENTS   OF   ALGEBRA. 

third  $275  more  than  the  second,  and  the  fourth  $389  more  than 
the  third.     Wliat  is  the  share  of  each  ? 

2.  Three  men  share  a  certain  sum  in  the  following  manner ; 
the  sum  of  A's  and  B's  shares  is  $123,  that  of  A's  and  C's  $110, 
and  that  of  B's  and  C's  $83.  What  is  the  whole  sum  and  the 
share  of  each  ? 

Let  X  =  the  whole  sum,  a,  h,  and  c  the  sum  of  the  shares  of 
A  and  B,  A  and  C,  B  and  C,  respectively;  then  x  —  a=C's 
share,  &c.,  and  we  have 

«  +  h-\-c 

74.  The  seventh  question,  art.  15,  may  be  stated  generally, 
thus.  A  cistern  is  supplied  by  two  pipes  ;  the  first  will  fill  itUn 
a  hours,  the  second  in  h  hours.  In  what  time  will  the  cistern  he 
filled  if  both  run  together  ? 

Let  X  =  the  time ;  the  capacity  of  the  cistern  being  supposed 
equal  to  unity,  we  have 

a^b  —  ^' 
whence  freeing  from  denominators 

ax-{-bx  =  ab. 
Here  it  will  be   observed,  that  x  is  taken  a  times  and  also  b 
times;  whence  on  the  whole  it  is  taken  a-\-b  times;   a-{-b 
is   then  the   coefficient  of  x,  and   the  above  equation  may  be 
written  thus, 

{a-{-h)  x  =  ab  ; 

.  ab 

whence  x  =  — r—: . 

a-f-b 

Translating  this  formula.  We  have  the  following  rule  foi 
every  case  of  the  proposed  question,  viz.  Divide  the  product 
of  the  numbers,  which  denote  the  times  employed  by  each  pipe  in 
flUing  the  cistern,  by  the  sum  of  these  numbers;  the  quotient  loill 
he  the  time  required  by  both  the  pipes  running  together  to  fill 
the  cistern. 


SOLUTION   OF    QUESTIONS   IN  A  GENERAL   MANNEE.  VtS 

Example.  Suppose  one  pipe  will  fill  the  cistern  in  5 J  hours, 
and  the  other  in  9  hours ;  in  what  time  will  it  be  filled  if  both 
run  together  ? 

75.  The  4th  question,  art.  6,  may  be  thus  generalized.  A 
gentleman  meeting  four  poor  persons  distributed  a  shillings  ammig 
them ;  to  the  second  he  gave  b  times^  to  the  third  c  times ^  and  to 
the  fourth  d  times  as  much  as  to  the  first.  What  did  he  give  to 
each"^ 

Let  X  repiresent  what  he  gave  to  the  first,  we  then  have 
z-\'bx-\'CX'\-  dx  =  a, 
or  {1  -\-  b  -{-  c  '\-  d)  x  =  a  ; 

whence  a;=:.,    ,    ,    , f— 3. 

\ -\- b -\- c -\- d 

Let  next  the  following  questions  be  generalized. 

1.  A  bankrupt  wishing  to  distribute  his  remaining  property 
among  his  creditors,  finds,  that  in  order  to  pay  them  $175  apiece, 
he  should  want  $30,  but  if  he  pays  them  $168  apiece  he  will 
have  $40  left.     How  many  creditors  had  he  ? 

2.  It  is  required  to  divide  the  number  91  into  two  such  parts, 
that  the  greater  being  divided  by  their  difference  the  quotient 
may  be  7. 

3.  Divide  the  number  138  into  two  such  parts,  that  5  times 
the  first  part  diminished  by  4  times  the  second  will  be  equal 
to  85. 

4.  Three  men,  A,  B,  and  C,  engage  in  trade  and  gain  $500, 
of  which  C  is  to  have  twice  as  much  as  B,  and  B  $50  less  than 
4  times  as  much  as  A.    How  much  will  each  receive  ? 

5.  A  trader  having  gained  $3450  by  his  business,  and  lost 
$2375  by  bad  debts,  found,  that  |  of  what  he  had  left  equalled 
the  capital  with  which  he  commenced  trade.  What  was  his 
capital? 

6.  In  a  certain  school  \  of  the  pupils  lean  nayigation,  |  leaia 


94  ELEMENTS   OF  ALGEBRA. 

geometry,  ^  learn  algebra,  and  the  rest,  23  in  number,  leam 
arithmetic.     How  many  pupils  are  there  in  all  ? 

76.  The  nineteenth  question,  art.  15,  may  be  presented  in  a 
general  manner,  thus.  A  laborer  was  hired  for  a  certain  number 
a  of  days;  for  each  day  that  he  wrought  he  loas  to  receive  b  shil' 
lings,  but  for  each  day  that  he  was  idle,  he  was  to  forfeit  c 
shillings.  At  the  end  of  the  time  he  received  d  shillings.  How 
many  days  did  he  loorh,  and  how  many  was  he  idle  ? 

Putting  X  =  the  number  of  days,  in  which  he  wrought,  and 
resolving  the  question,  we  obtain 

d-\-ac 
b-j-c 

Example.  A  laborer  was  hired  for  75  days ;  for  each  day  that 
he  wrought  he  was  to  receive  $3,  but  for  each  day  that  he  was 
idle,  he  was  to  forfeit  $7.  At  the  end  of  the  time  he  received 
$125.  To  determine  by  the  above  formula  the  number  of  days 
in  which  the  laborer  wrought. 

The  two  following  questions  may  also  be  solved  by  the  same 
formula.     Why  is  this  the  case  ? 

1.  A  man  agreed  to  carry  20  earthen  vessels  to  a  certain 
place  on  this  condition;  that  for  every  one  delivered  safe  he 
should  receive  11  cents,  and  for  every  one  he  broke,  he  should 
forfeit  13  cents;  he  received  124  cents.  How  many  did  he 
break  ? 

2.  A  fisherman  to  encourage  his  son  promises  him  9  cents 
for  each  throw  of  the  net  in  which  he  should  take  any  fish, 
but  the  son,  on  the  other  hand,  is  to  forfeit  5  cents  for  each 
unsuccessful  throw.  After  37  throws  the  son  receives  from  the 
father  235  cents.  What  was  the  number  of  successful  throws 
of  the  net? 

77.  Lei  it  be  proposed  next  to  make  a  rule  for  Fellowship,  and 
in  order  to  this,  let  us  take  the  following  example. 

Three  men,  A,  B,  and  C  commence  trade  together,  and  fur- 
nish money  in  proportion  to  the  numbers  w,  n  and  p  respectively; 


T 


SOLUTION    OF   QUESTIONS   IN  A  GENERAL   MANNER.  95 

they  gain  a  certain  sum  a.    What  is  each  man's  share  of  the 
gain  ? 

Let  X  =  A's  share ; 

7tX  7)  X 

then  —  =  B's,  and  —  =  C's  share. 
m  m 

By  the  question,  therefore, 

.  nx  ,  px 
x-\ \-'—  =  a. 

Freeing  from  denominators,  we  have 

mx-\-nx-\-'px-=.ma^ 
or,  which  is  the  same  thing 

{m -\- n -\- "p)  x=^ma; 

whence  x  = ; ; —  =  A's  share. 

m-f-n-f-p 

Muhiplying  next  the  value  of  x  by  w,  and  dividing  by  tw,  we 
obtain 

na 


m  ■\'n'\-p 
In  like  manner,  we  find 

pa 


=  B's  share. 


=  C's  share. 


m-\-n-\-p 

To  find  a  share  of  the  gain  therefore;  Multiply  the  corre- 
sponding proportion  of  the  stock  into  the  whole  gaiuy  and  divide 
the  product  by  the  sum  of  the  proportions. 

78.     Let  now  the  following  questions  be  generalized. 

1.  Three  merchants,  A,  B,  C,  enter  into  partnership.  A  ad- 
vances $750,  B  $1300  and  C  $825.  A  leaves  his  money  9 
months,  B  13  months,  and  C  15  months  in  the  business.  They 
gain  $830.    What  is  the  share  of  each  ? 

Since  A  advances  $750  for  9  months,  he  advances  what  is 
equivalent  to  $750  X  9  for  1  month.  In  like  manner  B  advances 
what  is  equivalent  to  $1300  X  13  for  one  month,  &c. 

Let  p,  p\  p"  represent  respectively  the  sums  advanced  by, 
each,  and  j,  j',  g",  the  times  in  which  these  sums  were  severally 


96  ELEMENTS    OF   ALGEBRA. 

employed ;  putting  a  for  the  sum  gained,  and  z  for  A's  share  of 
the  gain,  we  have  • 


3.  A  bankrupt  leaves  $18000  to  be  divided  among  three 
creditors,  in  proportion  to  their  claims.  Now  A's  claim  is  to  B's 
as  2  to  3,  and  B's  claim  to  C's  as  4  to  5.  How  much  does  each 
creditor  receive  ? 

3.  A  gentleman  hired  three  men  to  perform  a  certain  piece  of 
work ;  the  first  working  9  hours  a  day  would  perform  the  work 
in  10  days,  the  second  working  7  hours  a  day,  in  15,  and  the 
third,  working  12  hours  a  day,  in  6  days.  How  long  will  it  take 
them  together  to  perform  the  work  ? 

4.  A  merchant  purchased  24  yards  of  cloth  of  two  different 
kinds  for  $408.  The  first  cost  $18,  the  second  $15  a  yard. 
How  many  yards  were  there  of  each  kind  ? 

5.  A  gentleman  hired  two  workmen  for  50  days ;  to  the  first 
he  gave  $3,  to  the  second  $2  a  day.  On  settling  with  them  he 
paid  both  together  $130.     How  many  days  did  each  work  ? 

What  have  these  last  two  questions  in  common,  and  what 
general  statement  will  comprehend  both  ? 

79.  Thus  far  we  have  employed  the  first  letters  of  the  alphabet 
to  represent  known  quantities,  and  the  last  to  denote  those  which 
are  unknown.  In  some  cases  it  is  more  convenient  to  represent 
tlie  quantities,  whether  known  or  unknown,  by  the  initials  of  the 
words  for  which  they  stand. 

Let  it  be  proposed  to  determine  what  sum  of  money  must  be 
put  at  interest,  "at  a  given  rate,  in  order  to  amount  to  a  given  sum 
in  a  given  time.  • 

Let  p  ==  the  principal,  or  sum  put  at  interest, 
r  =  the  rate, 
fl  =  the  given  amount, 
;( =  the  given  time. 


SOLUTION   OF   QUESTIONS    IN    A   GENERAL    MANNER.  97 

By  the  question,  we  have  p-\-trp  =  a^ 
or  {l-\-tr)p  =  a; 

whence  p  =  - — ; . 

^       l-\-tr 

We  have  therefore  the  following  rule,  by  which  to  find  the 
principal  required,  viz.  Multiply  the  rate  by  the  time  and  add  1 
to  the  product ;  the  amount  divided  by  the  sum  thus  obtained  will 
give  the  principal. 

Examples.  1.  What  sum  of  money  must  be  put  at  interest 
at  6  per  cent.,  in  order  that  the  principal  and  interest  may,  at  the 
end  of  5  years,  amount  to  $748,80  ? 

Six  per  cent,  will  be  $6  on  $100,  or,  $.06  on  one  dollar ;  r  in 
the  formula  will  be  then,  for  this  case,  .06  and  we  obtain  $576 
for  the  answer. 

2.  A  man  lent  a  certain  sum  of  money  at  5  per  cent. ;  at  the 
end  of  7  years  he  received  for  principal  and  interest  $1237.47, 
W]>at  was  the  sum  lent  ?  Ans.  $916.65. 

3.  A  merchant  finds  that  by  a  fortuiaate  speculation  with  his 
floating  capital,  he  has  gained  15  per  cent.,  and  that  by  this 
means  it  has  increased  to  S15571.     What  was  his  capital? 

Ans.  $13540. 

80.  The  equation p-\-trp=^a,  contains,  it  will  be  perceived, 
four  different  things,  any  one  of  which  may  be  determined,  when 
the  others  are  known.  Deducing,  for  example,  the  value  of  ty 
we  have 

a  —  p 
rp 

Whence  to  find  the  time,  when  the  amount,  principal  and 
rate  are  given ;  From  the  amount  subtract  the  principal^  and 
divide  the  remainder  by  the  product  of  the  rate  multiplied  by  the 
principal. 

Examples.  1.  A  man  put  at  inJerest  $345  at  4  per  cent.; 
at  the  end  of  a  certain  time  he  received  for  principal  and  interest 

7 


^  ELEMENTS   OF   ALGEBRA. 

$483.     Required  the  time  for  which  the  money  was  lent. 

Ans.  10  years. 

2.  A  merchant  lets  out  his  floating  capital,  amounting-  to  $5873 
at  10  per  cent,  interest.  At  the  end  of  a  certain  number  of  years 
he  finds  that  he  has  received  $3523,80  interest.  For  how  many 
years  was  his  capital  let  out  ?  Ans.  6. 

3.  Let  the  learner  prepare  the  formula  and  solve  the  following 
example. 

A  gentleman  put  at  interest  $6840,  and  at  the  end  of  5  years 
received  for  capital  and  interest  $8208.  What  rate  per  cent,  did 
he  receive  ?  Ans.  4  per  cent. 

81.  In  the  preceding  questions  the  object  has  been  to  deter- 
mine certain  unknown  numbers  by  means  of  others,  which 
are  known,  and  which  have  relations  to  the  unknown  num- 
bers established  by  the  enunciation  of  the  question.  We  shall 
now  show  the  aid  derived  from  the  same  signs  in  demon- 
strating certain  properties  in  relation  to  known  and  given 
numbers. 

1.  To  demonstrate  that  if  both  terms  of  a  fraction  be  multi- 
plied by  the  same  number,  the  value  of  the  fraction  will  not  be 
changed. 

Let  the  proposed  fraction  be  designated  by  -,  and  let  n  be  any 

number  whatever. 

Putting  -  =  m,  we  have  a  =  bm; 

multiplying  both  sides  of  this  last  by  n,  we  have 

na=inbm, 
from  which  we  deduce 

na 

no 

,  na       a 

whence  -—=-. 

no       h 

2.  Let  the  same  number  be  added  to  both  terms  of  a  proper 


SOLUTION   OF    QUESTIONS   IN   A    GENERAL    MANNER.  99 

fraction;  to  determine  what  effect  this  will  produce  upon  the 
value  of  the  fraction. 

Let  us  designate  the  fraction  by  -.     Let  m  he  the  number 

added  to  both  terms  of  this  fraction;  it  then  becomes 

To  compare  the  two  fractions,  it  is  necessary  to  reduce  them 
to  the  same  denominator.  Performing  this  operation,  we  have 
for  the  first 

ab-\-am 


and  for  the  second 


bb  -\-bm 
ab-\-  bm 


bb-\-bm 

Here  the  two  numerators  have  the  part  ab  common  to  both; 
but  the  part  bm  oi  the  second  is  greater  than  the  part  am  oi  the 
first,  since  b  is  greater  than  a ;  the  second  fraction  is  therefore 
greater  than  the  first ;  whence,  If  the  same  number  be  added  to 
both  terms  of  a  proper  fraction^  the  value  of  the  fraction  will  be 
increased* 

3.  It  has  been  shown  in  arithmetic  that,  Every  divisor  common 
to  tivo  numbers  must  divide  the  remainder  after  the  division  of 
the  greater  of  these  numbers  by  the  less.  Let  us  now  demonstrate 
this  property  by  the  aid  of  algebraic  symbols. 

Let  D  be  the  divisor  common  to  the  two  numbers ;  let  A  D 
represent  the  greater  of  the  two  numbers  and  B  D  the  less ;  let 
Q  be  the  entire'  quotient  arising  from  the  division  of  the  greater 
by  the  less,  and  let  R  be  the  remainder ;  we  have  then 

AD  =  BDxQ+.R; 

dividing  both  sides  by  D,  we  have 

A  =  BxQ+5. 

Here  the  first  member  of  the  equation  is  an  entire  numli^r, 


100  ELEMENTS    OF   ALGEBRA. 

the  second  must,  therefore,  be  equal  to  an  entire  number;  but 
of  this  member  the  term  BQ  is  an  entire  number;  whence 

_  must  be  an  entire  number,  that  is,  R  must  be  exactly  divisible 

by  D.     The  proposition  above  is,  therefore,  demonstrated. 
The  following  propositions  may  now  be  demonstrated. 

1.  If  the  sum  of  any  two  quantities  be  added  to  their  differ- 
ence, the  sum  will  be  twice  the  greater. 

2.  If  the  difference  of  any  two  quantities  be  taken  from  their 
sum,  the  remainder  will  be  twice  the  less. 

3.  The  second  power  of  the  sum  of  two  quantities  contains 
the  second  power  of  the  first  quantity,  plus  double  the  pro- 
duct of  the  first  by  the  second,  plus  the  second  power  of  the 
second. 

4.  The  second  power  of  the  difference  of  two  quantities  is 
composed  of  the  second  power  of  the  first  quantity,  minus  the 
double  product  of  the  first  by  the  second,  plus  the  second  power 
of  the  second. 

5.  The  product  of  the  sum  and  difference  of  two  quantities  is 
equal  to  the  difference  of  their  second  powers. 

The  questions,  art.  15,  will  furnish  additional  exercises  for 
the  learner  in  stating  and  resolving  questions  in  a  general 
manner.  * 


SECTION  IX. — Discussion  of  Problems  and  Equations  op 
THE  First  Degree. 

82.  When  a  problem  has  been  solved  in^  a  general  manner, 
it  may  be  proposed  to  determine  what  values  the  unknown 
quantities  will  take  for  particular  hypotheses  made  upon  the 
known  quantities.    The  determination  of  these  different  values, 


DISCUSSION   OF  FORMI^LAS^       ]  ',  '   ','    ;  I.;    >  ,101' 

and  the  interpretation  of  the  results  to  which  we  arrive,  form 
what  is  called  the  discussion  of  the  problem. 

The  discussion  of  the  following  problem  presents  nearly  all 
the  circumstances,  that  can  ever  occur  in  equations  of  the  first 
degree. 

Two  couriers  set  out,  at  the  same  time,  from  two  different 
points  A  and  B,  in  the  line  E  D,  and  travel  towards  D  until 
they  meet ;  the  courier,  who  sets  out  from  the  point  A,  travels 
at  the  rate  of  m  miles  an  hour,  the  other  travels  at  the  rate  of 
n  miles  an  hour ;  the  distance  between  the  points  A  and  B  is 
a  miles;  at  what  distance  from  the  points  A  and  B  will  they 
meet  ? 

I  I 

E  C  A  B  C  D 

Suppose  C  to  be  the  point  in  which  they  meet ;  let  a;  =  the 
distance  A  C,  y  =  the  distance  B  C.  We  have  for  the  first 
equation  x  —  y  =  a. 

The  first  courier,  travelling  at  the  rate  of  m  miles  an  hour, 

cc 
will  be  —  hours  in  passing  over  the  distance  x;  the  second, 
m 

travelling  at  the  rate  of  n  miles  an  hour,  will  be  -  hours  in 

n 

passing  over  the  distance  y;   and  since  these  distances  must 

each  be  passed  over  in  the  same  time,  we  shall  have  for  the 

second  equation 

m      n 

Resolving  these  two  equations,  we  have 

am         an 

m  —  n  m  —  n 

Discussion, 

1.  Let  m  be  greater  than  n.     In  this  case  the  values  of  x  and 
y  will  be  positive,  and  the  problem  will  be  solved  in  the  exact 
of  the  enunciation  j  for,  it  is  evident,  that  if  the  courier, 
I* 


.tte  :  ^'l"*,.^'  ';  ;       '  p'^Eli'E^TS   OF   ALGEBRA. 

who  sets  out  from  A,  travels  fasti^r  tlian  the  other,  they  will 
meet  somewhere  in  the  direction  A  D. 

2.  Let  n  be  greater  than  m.     This  being  the  case  we  shall 
have 


am  an 

y 


n  —  m  n  —  m 

Here  the  values  of  x  and  y  are  negative.  In  order  to  interpret 
this  result,  we  observe  that  the  courier  from  B  travelling  faster 
than  the  courier  from  A,  the  interval  between  them  must  in- 
crease continually.  It  is  absurd  therefore  to  require  that  they 
should  meet  in  the  direction  A  D.  The  negative  values  for 
X  and  y  indicate,  then,  an  absurdity  in  the  conditions  of  the 
question.  To  show  how  this  absurdity  may  be  done  away,  let 
us  substitute  in  the  equations  of  the  problem  —  x,  —  y  instead 
of  X  and  y,  we  shall  then  have 

—  x-{-y=^a>i         (y  —  x  =  a 

—  —  =  — ^{   ^^  ]-=t 
m  n)         (m       n 

The  second  equation  is  not  affected  by  the  change  of  sign,  as 
indeed  it  ought  not  to  be,  since  it  only  expresses  the  equal- 
ity of  the  times.  In  regard  to  the  first,  however,  we  have 
y  —  a:  =  «,  instead  of  x  —  y=.a.  This  shows  that  the  point, 
in  which  the  couriers  meet,  must  be  nearer  to  A  than  to  B  by 
the  distance  A  B;  it  must,  therefore,  be  at  some  point  C  on 
the  other  side  of  A  with  respect  to  B.  In  order  then  to  remove 
the  absurdity  in  the  enunciation  of  the  question,  it  is  necessary 
to  suppose  the  couriers,  instead  of  travelling  in  the  direction 
A  D,  to  travel  in  the  opposite  direction  B  E.  Indeed,  if  we 
resolve  the  equations 

y  —  x  =  a 


we  have  a:  = ,  y  = ,  values  which  are  positive, 

n  —  m  n  —  m 

and  which  answer  the  conditions  of  the  probletri  modified,  thus, 

Two  couriers  set  out  at  the  same  time  from  two  points,  A  dfid 


DISCUSSION   OF    FORMULAS.  103 

B,  in  the  line  E  D,  and  travel  towards  E  ;  the  courier,  who  sets 
out  from  the  point  B,  travels  at  the  rate  of  n  miles  an  hour,  the 
other  travels  at  the  rate  of  m  miles  an  hour;  the  distance  be- 
tween the  points  B  and  A  is  a  miles ;  at  what  distance  from  the 
points  B  and  A  will  they  meet  ? 

3.  Let  m=.n.  In  this  case  we  have  m  —  n=0,  and  the 
values  of  x  and  y  become 

am  an 

But  how  shall  we  interpret  this  result  ?  Returning  to  the 
question,  we  perceive  it  to  be  absolutely  impossible  to  satisfy 
the  enunciation;  for,  the  couriers  travelling  equally  fast,  the 
interval  between  them  must  always  continue  the  same,  how- 
ever far  they  may  travel  in  either  direction.  It  is  impossible, 
then,  that  they  should  meet,  and  no  change  in  the  enunciation, 
so  long  as  we  have  m  =  n,  can  make  it  possible.  Indeed, 
the  equations  of  the  problem  on  the  hypothesis  m=.n  become 

X  —  y  =  a 

x  —  yz=0, 
equations,   which   are    evidently   incompatible.      Zero   being  a 
divisor  is,  then,  a  sign  of  impossibility. 

The  expressions  -^,  -^  are  considered,  however,  by  mathe- 
maticians as  forming  a  species  of  value  for  x  and  y,  to  which 
they  give  the  name  of  infinite  value.  To  show  the  reason  for 
this,  let  us  suppose  that  the  difference  between  m  and  7i  with- 
out being  absolutely  nothing  is  very  small;  in  this  case,  it  is 
evident  that  the  values  of  x  and  y  will  be  very  largo.  Let,  for 
example,  m  =  3,  m  —  tz  =  0. 01,  we  shall  then  have  n  =  2. 99, 
whence 

am    3  a _-^_  an 

m  —  n       .01  '  m 

Again  let  m  —  7z  =  .0001,  m  being  equal  to  3,  n  will  then 
=  2. 9999,  whence 

-^^  =  30000a,  — —  =  29999a. 
m  —  n  m  —  n 


104  ELEMENTS    OF    ALGEBRA. 

In  a  word,  so  long  as  there  is  any  difference,  however  small, 

between  m  and  n,  the  couriers  will  meet  in  one  direction  or 

the  other;  but  the  distance  of  the  point,  in  which  they  meet, 

from  the  points  A  and  B  will  be  greater  in  proportion  as  the 

difference  between  m  and  n  is  less.     If  then  the  difference  he- 

tween  m  and  n  is  less  than  any  assignable  quantity,  the  distances 

am         an         .^^  .  ^  .       ,, 
, will  be  greater  than  any  assignable  quantity  or 

infinite.  Since  then  0  is  less  than  any  assignable  quantity,  we 
may  employ  this  character  to  represent  the  ultimate  state  of  a 
quantity  which  may  be  decreased  at  pleasure ;  and  since  the 
value  of  a  fractional  quantity  is  greater,  in  proportion  as  its 

denominator  is  less,  the  expression  — r-,  and  in  general,  any 

quantity  with  zero  for  a  denominator  may  be  considered  as  the 
symbol  of  an  infinite  quantity,  that  is,  a  quantity  greater  than 
any,  which  can  be  assigned. 

We  say  then  that  the  values  x=--^,  y=.-—  are  infinite. 

To  show  how  the  notion  indicated  by  the  expression  -r--  does 

away  the  absurdity  of  the  equations 

X  —  y=-a,  X  —  2/  =  ^' 
from  the  second  of  these  equations,  we  deduce  the  value  of  y 
and  substitute  it  in  the  first,  we  then  have  x  —  x  =  a;  dividing 
both  sides  of  this  last  by  x,  we  have 

l_l=f,.or?  =  0. 
x         x 

Here,  as  we  put  for  x  values  greater  and  greater,  the  fraction 
-  will  differ  less  and  less  from  0,  and  the  equation  will  approach 
nearer  and  nearer  to  being  exact.  If  then  x  be  greater  than  any 
assignable  quantity,  -  will  be  less  than  any  assignable  quantity 
or  zero. 


DISCUSSION   OF    FORMULAS.  105 

4.  Let  us  suppose  next  m=:n,  and  at  the  same  time  a  =  0, 
we  shall  then  have 

0  0 

^  =  0'  2^  =  0- 

But  how  shall  we  interpret  this  new  result?  Returning  to 
the  enunciation,  we  perceive,  that  if  the  couriers  set  out  each 
from  the  same  point  and  travel  equally  fast,  there  is  no  par- 
ticular point  in  which  they  can  be  said  to  meet,  since  in  this 
case,  they  will  be  together  through  the  whole  extent  of  their 
route.  Indeed,  on  this  hypothesis  the  equations  of  the  problem 
become 

a:  — y  =  0, 

a;  — y  =  0, 
equations  which  are  identical ;  the  problem  is  tnerefore  indetev' 
mijiate,  since  we  have  in  fact  but  one  equation  with  two  un- 
known quantities.     The  expression  -  is  therefore  a  sign  of  inde-- 

termination  in  the  enunciation  of  the  problem. 

The  preceding  hypotheses  are  the  only  ones,  which  lead  to 
remarkable  results.  They  are  sufficient  to  show  the  manner 
in  which  algebra  corresponds  to  all  the  circumstances  m  the 
enunciation  of  a  problem. 

GENERAL   FORMULAS    FOR    EQUATIONS    OF   THE    FIRST   DEGREE   WITH 
ONE    OR   TWO   UNKNOWN    QUANTITIES.    . 

83.  Every  equation  of  the  first  degree  with  one  unknown 
quantity  may,  by  collecting  all  the  terms  which  involve  a:,  into 
one  member  and  the  known  quantities  into  the  other,  be  re- 
duced to  an  equation  of  the  form  Aa:=  B,  A  and  B  denoting 
any  quantities  whatever,  positive  or  negative. 

Let  there  be,  for  example,  the  equation 

mz 

p  =  x  —  a. 

n        ^  ^ 

Freeing  from  denominators,  transposing  and  upiting  terms,  we 

have  {m  —  n)x=.n{p  —  q). 


106  ELEMENTS    OF    ALGEBRA. 

Comparing  this  equation  with  the  general  formula,  we  have 
7»  —  7^  =  A,  n  {p  —  q)  =  'B. 

■p 
84.  Resolving  the  equation  Aa;  =  B,  we  have  «  =  -t-.     This 

A. 

is  a  general  solution  for  equations  of  the  first  degree,  with  one 
unknown  quantity. 

Discussion, 

1.  Let  it  be  supposed,  that  in  consequence  of  a  particular 
hypothesis  made  upon  the  known  quantities,  we  have  A  =  0, 

T> 

the  value  of  z  will  then  be  -jr.     But  the  equation  Aa;  =  B  on 

this  hypothesis  becomes  0  "^  x=B,  an  equation  which,  it  is 

evident,  cannot  be  satisfied   by  any  determinate   value  for  x. 

The   equation   0  X  ^  =  B   may,   however,   be   put    under   the 

■p 
form  — ■  =  0.     Here,  if  we  consider  x  greater  than  any  assign- 

■p 

able  quantity,  the  fraction  —  will  be  less  than  any  assignable 

quantity  or  zero.  On  this  account  we  say  that  infinity  in  this 
case  satisfies  the  equation.  It  is  evident,  at  least,  that  the  equa- 
tion cannot  be  satisfied  by  any  finite  value  for  x. 

2.  Let  us  suppose  next  A  =  0,  and  at  the  same  time  B  =  0, 

the  value  of  x  will  then  take  the  form  ^.  In  this  case  the  equa- 
tion becomes  0  X  ^  ==  0?  an  equation  which  may  be  satisfied  by 
any  finite  quantity  whatever,  positive  or  negative.  Thus  the 
equation,  or  the 'problem,  of  which  it  is  the  algebraic  translation, 
is  indeterminate. 

It  should  be  observed,  however,  that  the  symbol  *  does  not 

always  indicate  that  the  problem  is  indeterminate. 

Let,  for  example,  the  value  of  x  derived  from  the  solution  of  a 
problem  be 


DISCUSSION   OF   FORMULAS.  107 

If  we  put  a  =  b  in  this  formula,  it  will,  under  its  present 
form,  be  reduced  to  ^ ;  but  this  value  for  x  may  be  put  Tinder 
the  form 

^""        {a  —  b){a-^b)       ' 
If  then,  before  making  the  hypothesis  a  =  i,  we  suppress  the 
factor  a  —  bj  the  value  of  x  becomes 

a^  +  ab-^b\ 
a  +  b        ' 

from  which  we  obtain  x  =  —,  on  the  hypothesis  a  =  b. 

We  conclude  therefore  that  the  symbol  j-  is  sometimes  in  alge- 
bra the  sign  of  the  existence  of  a  factor  common  to  the  two  terms 
of  a  fraction,  ivhich  in  consequence  of  a  particular  hypothesis  be- 
comes 0,  and  reduces  the  fraction  to  this  form. 

Before  deciding  then,  that  the  result  tt  is  a  sign  that  the 

problem  is  indeterminate,  we  must  examine  whether  the  ex- 
pressions for  the  unknown  quantities,  which  in  consequence  of 
a  particular  hypothesis  are  reduced  to  this  form,  are  in  their 
lowest  terms,  if  not,  they  must  be  reduced  to  this  state ;  the  par- 
ticular hypothesis  being  then  made  anew,,  the  result  ^  shows 

that  the  problem  is  really  indeterminate. 

85.  Every  equation  of  the  first  degree  with  two  unknown 
quantities  may  be  reduced  to  an  equation  of  the  form 

A,  B,  and  C  denoting  any  quantities  whatever,  positive  or  nega- 
tive. It  is  evident,  that  all  equations  of  the  first  degree  with 
two  unknown  quantities  niay  be  reduced  to  this  state,  P.  by 
freeing  the  equation  from  denominators ;  2°.  by  collecting  into 
one  member  all  the  terms,  which  involve  x  and  y,  and  the 
known  quantities   into   the   other;    3**.   by  uniting  the  terms 


108  ELEMENTS    OF    ALGEBRA. 

which  contain  x  into  one  term,  and  those  which  contain  y  into 
another. 

Let  us  tak^  the  equations 

A'a;-j-BV  =  C'. 
Resolving  these  equations  we  have 

_CB^  — BC^      _AC^--CA/ 
"^""AB'  — BA"  ^""AB— BA'* 

This  is  a  general  solution  for  all  equations    of    the     first 
degree  with  two  unknown  quantities. 

To  show  the  use  which  may  be  made  of  these  formulas  in  the 
solution  of  equations,  let  there  be  the  two  equations, 
5a;4-32/=19,     4a:  +  7y  =  29. 
Comparing  these  with  the  general  equations,  we  have 
A  =  5,  B  =  3,  C  =  19,  A'  =  4,  B'  =  7,  C  =  29, 
whence,  by  substitution  in  the  formulas  for  x  and  y^  we  have 
19X7  —  8X29       133  —  87       46      ^ 


5X7  —  3X4          35—12 

-23*-"'^ 

5X29  —  19X4       145  —  76 
5X    7—   3X4        35  —  12 

-23-^' 

Discussion. 

In  the  above  formulas  for  x  and  ?/,  let  A  B'  —  B  A'  =  0, 
CB'  — BC  and  AC  — CA'  being  each  different  from  zero, 
we  shall  then  have 

CB'  — BC  AC  — CA' 

^  = "0 '      2/= 0 . 

To   interpret   these    results,   we   observe    that    the    equation 

AB' 

A  B'  —  B  A'  =  0  gives  A'  =  -r^r-  ;   substituting  this  value  in 

B 

the  equation  A'a:  -|-  B'y  =  C,  we  have 
A  VJ 


DISCUSSION   OF   FORMULAS.  109 

from  which  we  obtain  Aa;-|-By  =  -^;  comparing  this  with 

the  equation  Aa;  +  By  =  C,  the  left  hand  members,  it  will  bo 
perceived,  are  identical,  while  the  right  are  essentially  different ; 
for  if  in  the  numerator  CB'— BC,  CB'  be  greater  than  BC, 

C  will  be  greater  than  -^^;  and  if  C  B'  be  less  than  B  C,  C  will 

be  less  than  -^57-    We  conclude,  therefore,  that  the  two  equations 
B 

proposed  cannot  in  this  case  he  satisfied^  at  the  same  time,  by  any 

system  whatever  of  finite  valiies  for  x  and  y.     The   question 

therefore  in  ihis  case  is  impossible. 

Again,  let  us  suppose  A  B' —  B  A'=  0,  and  at  the  same  time, 

C  B' —  BC'=  0 ;  the  value  of  x  in  this  case  is  reduced  to  jr. 

To  interpret  this  result,  we  remark  that  the  equations  proposed 
may,  in  consequence  of  the  relation  A  B' —  B  A'=  0,  be  put 
ffider  the  form 

Aa;4-By==C 

Aa;+By  =  ^g7-, 

equations  which  are  identical,  since  from  the  relation  C  B' —  BC 

=  0,  we  have  -^^j-  =  C. 

JD 

In  order  then  to  resolve  the  problem,  we  have  in  fact  but  one 
equation  with  two  unknown  quantities ;  the  question  therefore  is 

indeterminate. 

B  A' 

Since  the  equation  A  B'  —  B  A'  =  0  gives  B'=  —r—  we  have 

A. 

by  substitution  in  the  equation  C  B'  —  BC'  =  0. 

£J^-BC'=0, 

or  reducing,  AC  —  C  A'  =  0 ;  we  infer ^  therefore,  that  if  the 

valice  of  X  be  of  the  form  ^  the.  valtie  of  y  will  he  of  the  sarM 
form  and  the  converse. 

J 


tlO  ELEMENTS    OF   ALGEBRA. 


PROBLEMS   FOR    SOLUTION   AND   DISCUSSION. 

1.  A  mercharkt  has  two  sorts  of  wine,  one  of  which  costs  a, 
the  other  b  shillings  a  gallon;  from  these  he  would  make  a 
mixture  of  c  gallons  to  be  worth  d  shillings  a  gallon.  How 
much  of  each  must  he  take  ? 

Let  a:  =  the  number  of  gallons  of  the  first,  y  of  the  second, 

.  cid  —  b)  cia  —  d) 

we  nave  '   x  =  — ~^y  =  — — i. 

a  —  b      ^  a  —  b 

How  shall  we  interpret  these  results  1°.  when  i  or  a  is  equal 
to  d;  2°.  when  a^^b ;  2°.  when  az=b,  and  at  the  same  time, 
h=-d  ;  4°.  what  condition  is  necessary  in  order  that  the  question 
may  be  solved  in  the  exact  sense  of  the  enunciation? 

Ans.  In  the  first  case,  the  quantity  of  one  of  the  ingredients 
v/ill  be  0,  as  it  should  be,  since,  if  the  price  of  one  of  the  ingre- 
dients is  equal  to  that  of  the  mixture,  none  of  the  other  will  be 
needed  to  make  the  mixture  of  the  required  price.  In  the  second, 
since  the  prices  of  the  ingredients  are  both  the  same,  a  mixture 
of  a  different  price  cannot,  it  is  evident,  be  made  from  them ;  the 
question,  therefore,  requires  an  impossibility.  In  the  third  case, 
the  price  of  the  ingredients  and  that  of  the  mixture  being  each 
the  same,  whatever  number  of  gallons  be  taken  of  either,  the 
mixture  will  be  of  the  required  price ;  the  question  is,  therefore, 
indeterminate.  The  number  of  solutions  is,  however,  limited  by 
the  circumstance  that  the  number  of  gallons  of  both  ingredients 
taken  together  must  be  equal  to  c.  Finally,  in  order  that  the 
question  may  be  answered  in  the  exact  sense  of  the  enunciation, 
the  price  of  the  mixture  must  be  comprised  between  the  prices 
of  the  ingredients. 

2.  To  find  a  number  such,  that  if  it  be  added  to  the  num- 
bers a  and  b  respectively,  the  first  sum  will  be  m  times  the 
second. 

Putting  z  for  the  number,  we  have  x  =  -:; . 

1  —  m 


DISCUSSION   OF    FORMULAS.  Ill 

'How  shall  we  interpret  this  result  when  7w=  1?  How  when 
m=l,  and  at  the  same  time  a  =  b?  How  when  m  is  greater 
than  1,  and  mb  greater  than  a?  What  conditions  are  necessary, 
in  order  that  the  question  may  be  solved  in  the  exact  sense  of  the 
enunciation  ? 

3.  The  sum  of  two  numbers  is  a,  and  the  sum  of  their  pro- 
ducts by  the  numbers  m  and  n  respectively  is  b.  "WTiat  are  the 
numbers  ? 

Putting  X  and  y  for  the  numbers,  we  have 

b  —  na  ma  —  b 

x  = ,    7J. 


Tfi  —•—  n 

How  shall  we  interpret  these  results,  when  w  is  greater  than 

n  and  na  greater  than  b?     How  when  m  =  n?    How  Avhen 

m  =  n,  and  at  the  same  time  na  =  b?    What  conditions  are 

necessary  in  order  that  the  question  may  be  solved  in  the  exact 

sense  of  the  enunciation  ? 

4.  Two  numbers  are  in  proportion  of  fl  to  i  ;  but  if  c  be  added 

to  both,  they  will  then  be  in  proportion  of  m  to  n  ;  what  are  the 

numbers  ?  ^ 

.        acim  —  n)       .  bcim  —  n) 

Ans.  — ^^ ; — -  and  — ^ -, — -. 

an  —  bm  an  —  om 


SECTION  X.— Theory  of  Inequalities. 

86.  In  the  reasonings,  which  relate  to  the  discussion  of  a 
problem,  we  have  frequent  occasion  to  make  use  of  the  expres- 
sions '^greater  ihari^''  ^^less  than.'''  In  such  cases  we  shall  attain 
a  greater  degree  of  conciseness,  by  representing  each  of  these 
expressions  by  a  convenient  sign.  It  is  agreed  to  represent  the 
expression  "  greater  tkarC'  by  the  sign  ]>  ;  thus,  a  greater  than  h 
is  expressed  hy  g.^b.    The  same  sign  by  a  change  of  position 


112  ELEMENTS   OF   ALGEBRA. 

is  made  to  represent  the  phrase  "less  than;"  thus,  2  less  than  h 
is  expressed  by  a  <^  b. 

An  equation  of  the  form  a=za  is  called  an  eqality.  An 
expression  of  the  form  a  ]>•  3,  or  a  <^  ^  is  called  an  inequality. 

The  principles  established  for  equations  apply  in  general  to 
inequalities.  As  there  are  some  exceptions,  however,  we  shall 
state  the  principal  transformatioi^s,  which  may  be  made  upon 
inequalities,  together  with  the  exceptions  which  occur. 

1°.  We  may  always  add  the  same  quantity  to  both  members  of 
an  inequality,  or  subtract  the  same  quantity  from  both  members^ 
and  the  inequality  will  continue  in  the  same  sense. 

Thus-,  let  3  <^  5 ;  adding  8  to  both  sides,  we  have 
8-|-3<5  +  8,  or  11<13. 

Again  let  —  3  ^  —  5 ;  adding  8  to  both  sides  we  have 
8_3>8  — 5,  or5>3. 

This  principle  enables  us,  as  in  the  case  of  equations,  to 
transpose  a  term  from  one  member  of  an  inequality  to  the 
other;  thus,  from  the  inequality  a'^-)- ^^^^^^^^  —  aSwe  obtain 
2a«  +  i'^>3c* 

2°.  We  may  in  all  cases  add,  member  to  merriber,  two  or  more 
inequalities  established  in  the  same  sense,  and  the  in£qualityt 
which  results,  will  exist  in  the  sense  of  the  proposed. 

Thus,  let  there  he  a'^  b,  c"^  d,  e'^f;  we  have 
a  +  c  +  e>b  +  d+f 

But  ifive  subtract,  member  from  member,  two  or  more  inequal' 
ities  established  in  the  same  sense,  the  inequality,  which  results^ 
will  not  always  exist  in  the  sense  of  the  proposed. 

Let  there  be  the  inequalities  4  <^  7,  2  <]  3,  we  have  bysub- 
traction4  — 2<7  — 3,  or2<4. 

But  let  there  be  the  inequalities  9  <^  10  and  6  <  8,  subtract- 
ing the  latter  from  the  former,  we  have 

9  — 6>10  — 8,  or3>2. 

3*.  We  may  multiply  or  divide  the  two  members  of  an  inequal' 


THEORY  OP   INEQTTALITIES.  113 

ity  by  any  positive  or  absolute  number,  and  the  inequality ^  which 
results,  ivill  exist  in  the  sense  of  the  proposed. 

Thus,  if  we  have  a<^b,  multiplying  both  sides  by  5,  we  have 
5a<:^5b. 

By  means  of  this  principle,  we  may  free  an  inequality  from 
its  denominators      Thus,  let  there  be 

a^  —  b^a'  —  b^ 
2d    ^     3a    ' 
we  have  by  multiplication  (a''  —  b^)2a'^{a^  —  b^)2d,  and  by 
division  3a  ^2^. 

But  if  we  multiply  or  divide  the  two  members  of  an  inequality 
by  a  negative  quaiitity,  the  inequality,  which  results,  will  exist  in 
the  contrary  sense. 

Thus,  let  8  ^  7 ;  multiplying  both  sides  by  —  3,  we  have 
_24<  — 21. 

From  this  it  follows,  that  if  we  change  the  sign  of  each  term 
of  an  inequality,  the  inequality,  which  results,  will  exist  in  a  sense 
contrary  to  that  of  the  proposed ;  for  this  transformation  will  be 
equivalent  to  nntUiplying  both  members  by  —  1, 

87.  Let  there  now  be  proposed  the  inequality 

Multiplying  both  sides  by  3,  we  have 

21a:  — 23>2a:+15; 
whence  transposing  and  reducing,  we  have 

a:>2. 

Here  2  is  the  limit  to  the  value  of  x,  that  is,  if  we  substitute 
for  X  in  the  proposed  any  value  greater  than  2,  the  inequality 
will  be  satisfied.  The  process,  by  which  the  limit  to  the  value 
of  the  unknown  quantity  is  determined,  is  called  resolving  the 
inequality. 

8 


114  ELEMENTS    OF   ALGEBRA. 


EXAMPLES. 

1.  To  find  the  limit  to  the  value  of  x  in  the  inequalities 

2.  To  find  the  limit  to  the  value  of  x  in  the  inequalities 

X       X  ^6        X 

3.  To  find  the  limit  to  the  value  of  x  in  the  inequalities 

ax-\-  ah  <C—, 

88.  The  theory  of  inequalities  may  be  applied  to  the  solution 
of  certain  problems. 

1.  The  double  of  a  number  diminished  by  5  is  greater  than 
25,  and  triple  the  number  diminished  by  7  is  less  than  double  the 
number  increased  by  13.  Kequired  a  number  that  shall  possess 
tliese  properties. 

By  the  question,  we  have 

2a:  — 5>25 
3a;  — 7<2a;  +  13. 
Resolving  these  inequalities,  we  have  a;^  15,  a;<^20.     Any 
number  J^herefore,  entire  or  fractional,  comprised  between  15  and 
20  will  satisfy  the  conditions  of  the  question. 

2.  A  shepherd  being  asked  the  number  of  his  sheep  re- 
plied, that  double  their  number  diminished  by  7  is  greater 
than  29,  and  triple  their  number  diminished  by  5  is  less  than 
double  their  number  increased  by  16.  Required  the  number  of 
sheep. 


EXTRACTION  OF  THE  SQUARE  ROOT.  115 

Resolving  the  question,  we  have  x^  18,  and  a;<^21.  Here 
all  the  numbers,  comprised  between  18  and  21,  will  satisfy  the 
inequalities ;  but  since  the  nature  of  the  question  requires  that 
the  answer  should  be  an  entire  number,  the  number  of  solutions 
is  limited  to  2,  viz.  a;  =  19,  a;  =  20. 

3.  A  market  woman  has  a  number  of  oranges,  such,  that 
triple  the  number  increased  by  2,  exceeds  double  the  number 
increased  by  61 ;  and  5  times  the  number  diminished  by  70  is 
less  than  4  times  the  number  diminished  by  9.  How  many 
oranges  had  she  ? 

4.  The  sum  of  two  numbers  is  32,  and  if  the  greater  be 
divided  by  the  less,  the  quotient  will  be  less  than  5  but  greater 
than  2.     What  are  the  numbers  ? 

5.  The  sum  of  two  numbers  is  25 ;  if  the  greater  be  divided 
by  the  less,  the  quotient  will  be  greater  than  3,  and  if  the  less 
be  divided  by  the  greater  the  quotient  will  be  greater  than  \. 
What  are  the  numbers  ? 


SECTION  XI. — Extraction  of  the   Square  Root. 

89.  Let  it  now  be  proposed  to  find  a  number,  which  multiplied 
by  five  times  itself,  will  give  a  product  equal  to  te5. 

Putting  X  for  the  number  required,  we  have  by  the  question 
5r*=«:125,  from  which  we  obtain  x^  =  25.  This  equation  is 
essentially  different  from  any,  which  we  have  hitherto  considered. 
It  is  called  an  equation  of  the  second  degree,  because  it  contains 
X  raised  to  the  second  power.  To  find  the  value  of  x,  we  must 
see  what  number  ihultiplied  by  itself  will  give  25.  It  is  obvious, 
that  the  number  5  will  fulfil  this  condition ;  we  have  therefore 
«  =  5. 

The  value  of  x  is  easily  found  in  the  present  example,  but 


116  ELEMENTS    OF    ALGEBRA. 

in  Others  it  will  be  more  difficult.  Hence  arises  this  new  arith- 
metical question,  viz.  To  find  a  number,  which  multiplied  by 
itself  will  give  a  product  equal  to  a  proposed  number,  or  which 
is  the  same  thing,  from  the  second  power  of  a  number  to  deter- 
mine the  first. 

A  number,  which  multiplied  by  itself  will  produce  a  given 
number,  is  called  the  square  or  second  root  of  this  number.  The  • 
process  for  finding  the  second  root  is  called  extract^g  the  square 
or  second  root. 

In  the  following  table,  we  have  the  nine  primitive  numbers 
with  their  squares  written  under  them  respectively. 
1,    2,    3,    4,    5,    6,    7,    8,     9. 
1,    4,    9,  16,  25,  36,  49,  64,  81. 

By  inspection  of  this  table,  it  will  be  perceived,  that  among 
entire  numbers  consisting  of  one  or  two  figures,  there  are  nine 
only,  which  are  squares  of  other  entire  numbers.  The  remain- 
der have  for  a  root  an  entire  number  plus  a  fraction.  Thus  53, 
which  is  comprised  between  49  and  64,  has  for  its  square  root  7 
plus  a  fraction. 

The  numbers  in  the  second  line  of  this  table  being  the  squares 
of  those  in  the  first,  conversely,  the  numbers  in  the  first  line 
are  the  square  roots  of  those  in  the  second.  If,  therefore, 
the  number,  the  square  root  of  which  is  required,  consists  of 
one  or  two  figures  only,  its  root  will  be  readily  found  by  means 
of  the  table. 

Let  it  be  propose^  to  find  the  root  of  a  number  consisting  of 
more  than  two  figures,  6084,  for  example. 

The  square  of  9,  the  largest  number  consisting  of  one  figure, 
is  81,  and  the  square  of  100,  the  smallest  number  consisting  of 
three  figures,  is  10000 ;  the  square  root  of  6084  will,  therefore, 
consist  of  two  places,  viz.  units  and  tens. 

To  determine  then  a  method,  by  which  to  return  from  the 
proposed  number  to  its  root,  let  us  observe  the  manner,  in 
which  the  different  parts  of  a  number  consisting  of  two  places, 
47,  for  example,  are  employed  in  forming  the  square  of  this 


EXTRACTION  OF  THE  SQUARE  ROOT.  117 

number.  For  this  purpose  we  decompose  47  into  two  parts, 
viz.  40  and  7,  or  4  tens  and  7  units.  Designating  the  tens  by 
a  and  the  units  by  3,  we  have  a  -}"  ^  =  ^7,  and  squaring  both 
sides  a^-f-2a3-f-Z''^  =  2209.  Thus  the  square  of  a  number, 
consisting  of  units  and  tens,  is  composed  of  three  parts,  viz.  ike 
square  of  the  tens,  plus  twice  the  product  of  the  tens  multiplied, 
by  the  units,  plus  the  square  of  the  units.  Thus  in  2209,  the 
square  of  47,  we  have 

The  square  of  the  tens  {c^)  =  1600 

Tioice  the  tens  by  the  units  (2  a  3)  =   560 
The  square  of  the  units  {b^)  =     49 


2209 
Considering,  then,  the  proposed  number  6084  as  composed 
of  the  square  of  the  tens  of  the  root  sought,  twice  the  product 
of  the  tens  by  the  units,  and  the  square  of  the  units,  if  we  can 
discover  in  this  number  the  first  of  these  parts,  viz.  the  square 
of  the  tens,  the  tens  of  the  root  will  be  readily  found.  The 
square  of  an  exact  number  of  tens,  it  is  evident,  can  have  no 
figure  inferior  to  hundreds.  Separating  then  the  two  last  figures 
of  the  proposed  from  the  rest  by  a  comma,  the  square  of  the 
tens  will  be  found  in  60,  the  part  at  the  left  of  the  comma, 
which,  in  addition  to  the  hundreds  in  the  square  of  the  tens, 
will  also  contain  those,  which  arise  from  the  other  parts  of  the 
square.  60  is  comprised  between  49  and  64,  the  roots  of 
which  are  7  and  8  respectively ;  7  will,  therefore,  be  the  figure 
denoting  the  tens  in  the  root  sought.  Indeed  60  00  is  com- 
prised between  49  00,  and  64  00,  the  squares  of  70  and  80 
respectively ;  the  same  is  the  case  with  60  84 ;  the  root  required 
will  therefore  consist  of  7  tens  and  a  certain  number  of  units 
less  than  ten. 

The  figure  7  being  thus  obtained,  we  place  it  at  the  right 
of  the  proposed,  taking  care  to  separate  them  by  a  vertical 
line;  we  then  subtract  49,  the  square  of  7,  from  60,  and  to 
the  remainder  11  we  bring  down  84,  the  two  other  figures  of 


118  ELEMENTS    OF    ALGEBKA. 

the  proposed.  The  result  1184  of  this  operation  will  then  con- 
tain twice  the  product  of  the  tens  of  "the  root  hy  the  units,  plus 
th6  square  of  the  units.  Twice  the  product  of  the  tens  by  the 
units  will,  it  is  evident,  contain  no  figure  inferior  to  tens.  Sep- 
arating then  4,  the  right  hand  figure  of  the  remainder  1184, 
from  the  rest  by  a  comma,  the  part  118  of  this  remainder,  at  the 
left  of  the  comma,  must  contain  the  double  product  of  the  tens 
by  the  units,  together  with  the  tens  arismg  from  the  square  of 
the  units. 

The  double  product  of  the  tens  is  14;  dividing,  therefore, 
118  by  14,  the  quotient  8  will  be  the  unit  figure  exactly,  or  in 
consequence  of  the  tens  arising  from  the  square  of  the  units, 
it  may  be  too  large  by  1  or  2.  To  determine  whether  8  be  the 
right  figure  for  the  units  of  the  root,  we  multiply  twice  the  tens 
by  8  and  subtract  the  result  from  1184,  the  remainder  64  being 
equal  to  the  square  bf  8,  shows  that  8  is  the  unit  figure  sought. 
We  have  78,  therefore,  for  the  root  required.  The  operation 
will  stand  thus, 

60,84 
49 


118,4 
112 


14 

8 


64 
64 

To  complete  the  root,  we  place  8,  the  unit  figure,  at  the  right 
of  7,  the  figure  for  the  tens.  The  work,  moreover,  may  be 
abridged  by  writing  the  8  at  the  right  of  the  divisor,  and 
multiplying  148  the  number  thus  formed  by  8.  We  thus  ob- 
tain in  one  expression  twice  the  tens  by  the  units  and  the 
square  of  the  units;  this  being  equal  to  the  remainder  1184 
proves,  as  before,  that  8  is  the  right  figure  for  the  units  of  the 
root. 


EXTRACTION  OF  THE  SQUARE  ROOT.  119 

With  this  modification,  the  work  will  stand  thus, 

60, 84  I  78 
49 

118,4     148 
118  4 


Let  us  take,  as  a  second  example,  the  number  841.  Pursuing 
the  same  course  as  in  the  preceding  example,  we  find  2  for 
the  tens  of  the  root ;  subtracting  the  square  of  the  tens,  the 
remainder  will  be  441.  Separating  the  unit  figure  in  this  re- 
mainder from  the  rest  by  a  comma,  and  dividing  the  part  at  the 
left  by  double  the  tens,  in  order  to  obtain  the  unit  figure  of  the 
root,  we  have  11  for  the  result.  This  is  evidently  too  much. 
Indeed,  we  cannot  have  more  than  9  for  the  units ;  we  therefore 
try  9.  This  proves  to  be  the  correct  figure.  The  root  sought  is 
therefore  29. 

The  operation  will  be  as  follows  : 


8,41 
4 


29 


44,1  I  49 
441 


90.  Any  number  however  large  may  be  considered  as  com- 
posed of  units  and  tens ;  345,  for  example,  may  be  considered  as 
composed  of  34  tens  and  5  units. 

Let  it  now  be  proposed  to  find  the  second  root  of  190969. 
This  number  exceeds  10  000  and  is  less  than  1000  000 ;  its 
root  will  therefore  consist  of  three  places.  But  from  what  has 
been  said,  the  root  may  be  considered  as  composed  of  two 
parts,  units  and  tens.  The  proposed  will,  therefore,  consist 
of  three  parts,  viz.  the  square  of  the  tens  of  the  root,  twice 
the  tens  by  the  units  and  the  square  of  the  units.  The  square 
of  the  tens  will  have  no  figure  inferior  to  hundreds.  Separating, 
therefore,  the  last  two  figures  from  the  rest  by  a  comma,  the 
tens  of  the  root  will   be  found  by  extracting  the  square  root 


Sfe 


ELEMENTS    OF    ALGEBRA. 


of  1909,  the  part  of  the  proposed  at  the  left  of  the  comma. 
Regarding  1909  for  the  moment  as  a  separate  number,  its  root 
will  evidently  consist  of  two  places,  units  and  tens.  The  method 
of  finding  the  root  will,  therefore,  be  the  same  as  in  the  pre- 
ceding examples.  Performing  the  necossary  operations  we  obtain 
43  for  the  root  and  a  remainder  of  60.  There  will  therefore, 
be  43  tens  in  the  root  of  the  proposed,  and  bringing  down  the 
last  two  figures  of  the  proposed  by  the  side  of  60,  the  result 
6069  will  contain  twice  the  product  of  the  tens  of  the  root 
sought  by  the  units,  plus  the  square  of  the  units.  Separating, 
therefore,  the  right  hand  figure  from  the  rest  by  a  comma,  we 
divide  606,  the  part  on  the  left  of  the  comma,  by  86,  twice  the 
tens ;  this  gives  7  for  the  unit  figure.  Placing  the  7,  therefore, 
at  the  right  of  43,  the  part  of  the  root  already  found,  and  also 
at  the  right  of  86,  and  multiplying  this  last  by  7,  we  have  6069 
for  the  result.  7  is,  therefore,  the  right  unit  figure,  and  the  root 
of  the  proposed  is  437. 

The  following  is  a  table  of  the  operations. 

19,09,69  I  437 
16 


309  I  83 
249 

606,9 
606  9 


867 


The  same  process,  it  is  easy  to  see,  may  be  extended  to  any 
number  however  large.  From  what  has  been  done,  therefore, 
the  following  rule  for  the  extraction  of  the  second  root  will 
be  readily  inferred,  viz.  1°.  Separate  the  number  into  parts 
of  two  figures  each,  beginning  at  the  right.  2°.  Find  the  great- 
est second  power  in  the  left  hand  part ;  write  the  root  as  a  quO' 
tient  in  divisio?i,  and  subtract  the  second  power  from  the  left  hand 
part.  3°.  Bring  down  the  two  next  figures  at  the  right  of  the 
remainder  for  a  dividend  and  double  the  root  already  found  for 
a  divisor.     See  how  many  times  the  divisor  is  contained  in  the 


EXTRACTION  OF  THE  SQUARE  ROOT.  121 

dividend,  neglecting  the  right  hand  figure.  Write  the  result  in 
the  root  at  the  right  of  the  figure  previously  found,  and  also  at 
the  right  of  the  divisor.  4°.  Multiply  the  divisor,  thus  aug- 
mented, by  the  last  figure  of  the  root  and  subtract  the  product 
from  the  whole  dividend.  5°.  Bring  down  the  next  two  figures 
as  before,  to  form  a  neio  dividend,  and  double  the  root  already 
found  for  a  divisor,  and  proceed  as  before.  The  root  will  be 
doubled,  if  the  right  hand  figure  of  the  last  divisor  be  doubled. 

91.  If  the  number  proposed  be  not  a  perfect  square,  we  shall 
obtain  by  the  above  rule,  the  root  of  the  greatest  square  number 
contained  in  the  proposed.  Thus,  let  it  be  required  to  find  the 
square  root  of  1287.  Applying  the  rule  to  this  number,  we  ob- 
tain 35  for  the  root  with  a  remainder  62.  This  remainder  shows 
that  1287  is  not  a  perfect  square.  The  square  of  35  is  1225, 
that  of  36  is  1296 ;  whence  35  is  the  root  of  the  greatest  square 
contained  in  the  proposed. 

92.  When  the  proposed  number  is  not  a  perfect  square  a 
doubt  may  sometimes  arise,  whether  the  root  found  be  that  of 
the  greatest  square  contained  in  this  number.  This  may  be 
readily  determined  by  the  following  rule.  The  square  oi  a-\-\ 
h  c^  -\- 2a -{-!',  whence  the  square  of  a  quantity  greater  by 
unity  than  a  exceeds  the  square  oi  a  hj  2a-\-\.  From  this  it 
follows,  that  if  the  root  obtained  should  be  augmented  by  unity 
or  more  than  unity,  the  remainder  after  the  operation  must  be  at 
least  equal  to  twice  the  root  plus  unity.  When  this  is  not  the 
case,  the  root  obtained  is  that  of  the  greatest  square  contained  in 
the  proposed. 

EXAMPLES. 

1.  To  find  the  square  root  of  56821444.  Ans.  7538. 

2.  To  find  the  square  root  of  17698849.  Ans.  4207. 

3.  To  find  the  square  root  of  1607448649.         Ans.  40093. 

4.  To  find  the  square  root  of  12103441.  Ans.  3479. 

5.  To  find  the  square  root  of  48303584206084.  Ans.  6950078. 


122  ELEMENTS    OF    ALGEBRA. 

93.  From  what  has  heen  done,  it  will  be  perceived,  that  there 
are  many  whole  numbers,  the'  roots  of  which  are  not  whole 
numbers.  What  is  remarkable  in  regard  to  these  numbers  is, 
that  they  will  have  no  assignable  roots.  Thus  the  numbers 
3,  7,  11  have  no  assignable  roots,  that  is,  no  number  can  be 
found  either  among  whole  or  fractional  numbers,  which  multi- 
plied by  itself  will  produce  either  of  these  numbers.  The  proof 
of  this  depends  upon  the  following  proposition,  which,  we  shall 
now  demonstrate,  viz. 

Every  number  P,  which  luill  exactly  divide  the  product  A  B  q/* 
two  numbers  A  and  B,  and  which  is  prime  to  one  of  these  num- 
bers must  necessarily  divide  the  other  number. 

Let  us  suppose  that  P  will  not  divide  A,  and  that  A  is  greater 
than  P.  Let  us  apply  to  A  and  P  the  process  of  the  great- 
est common  divisor,  designating  the  quotients,  which  arise,  by 
Q>  Q'>  Q"  •  •  •  and  the  remainders  by  R,  R',  R"  .  .  .  respec- 
tively. It  is  evident,  that  if  the  operation  be  pursued  sufficiently 
far,  we  shall  obtain  a  remainder  equal  to  unity,  since  by  hypothe- 
sis A  and  P  are  prime  to  each  other.  .  This  being  premised  we 
have  the  following  equations 

A  =  PQ  +  R 
P  =  RQ'4-R' 
R  =  R'Q"+R" 


Multiplying  the  first  of  these  equations  by  B,  and  dividing 
by  P,  we  have 

-p-  =  BQ+-p- 

AB 

By  hypothesis  -p-  is  an  entire  number,  and  since  B  and  Q 

are  each  entire  numbers  the  product  BQ  is  an  entire  number. 

BR 

It  follows  therefore,  that  -p-  must  be  an  entire  number ;  whence 

B  multiplied  by  the  remainder  R  is  divisible  by  P. 


EXTRACTION  OF  THE  SQUARE  ROOT.  123 

Again,  multiplying  the  second  of  the  ahove  equations  by  B 
and  dividing  by  P,  we  have 

BRQ'   ,   BR' 


B 


p      -T-    p 


"RR 

But  we  have  already  shown  that  -5-  is  an  entire  number, 

whence  — p —  is  an  entire  number.     This  being  the  case,  -p- 

must  be  an  entire  number ;  whence  B  multiplied  by  the  remain- 
der R'  must  be  divisible  by  P. 

If  then  the  remainder  R'  is  equal  to  unity,  the  proposition 
is  demonstrated,  since  in  this  case  we  shall  have  B  X  1  or  B 
divisible  by  P.  But  if  the  remainder  R'  is  not  equal  to 
unity,  it  is  evident,  that  if  the  process  of  the  greatest  common 
divisor  be  applied  to  the  quantities  A  and  P  until  a  remain- 
der is  obtained  equal  to  unity,  we  may  in  the  same  manner 
as  above,  prove  that  B  multiplied  by  this  remainder  will  be 
divisible  by  P. 

We  conclude,  therefore,  that  if  P,  which  we  have  supposed 
not  to  divide  A,  will  not  divide  B,  it  will  not  divide  AB  the 
product  of  A  by  B. 

Returning  now  to  our  purpose,  it  is  evident,  in  order  that  a 

fractional  number  -  may  be  the  root  of  an  entire  number  c,  we 
must  have 

1^  =  '' 
But  if  c  be  not  a  perfect  square,  its  root  will  not  be  an  en- 
tire number,  that  is,  a  will  not  be  divisible  by  h;  but  from 
what    has   just    been  demonstrated,    if  a    is    not    divisible    by 
b,  ay^a  or  c^  will  not  be  divisible  by  3,  and  by  consequence 

c^  will  not  be  divisible  by  b^ ;  whence  —^  cannot  be  equal  to  an 

entire  number  c. 

94.  Though  the  roots  of  numbers,  which  are  not  perfect 
squares  cannot  be  assigned  either  among  whole  or  fractional 


124  ELEMENTS    OF    ALGEBRA. 

numbers,  yet,  it  is  evident,  there  must  be  a  quantity,  which  mul- 
tiplied by  itself  will  produce  any  number  whatever.  Thus  the 
root  of  53  cannot  be  assigned;  yet  there  must  be  a  quantity, 
which  multiplied  by  itself  will  produce  53.  This  quantity,  it  is 
evident,  lies  between  the  numbers  7  and  8,  for  the  square  of  7  is 
49,  and  the  square  of  8  is  64.  If  then  we  divide  the  difference 
between  7  and  8  by  means  of  fractions,  we  shall  obtain  numbers, 
the  squares  of  which  will  be  greater  than  49  and  less  than  64, 
and  which  will  approach  nearer  and  nearer  to  53. 

95.  All  numbers,  whether  entire  or  fractional,  have  a  common 
measure  with  unity;  on  this  account  they  are  said  to  be  cotw- 
mensurable ;  and  since  the  ratio  of  these  numbers  to  unity  may 
always  be  expressed  by  entire  numbers,  they  are  on  this  account 
called  rational  numbers. 

The  root  of  a  number  which  is  not  a  perfect  square  can  have 
no  common  measure  with  unity;  for,  since  it  is  impossible  to 
express  this  root  by  any  fraction,  into  how  many  parts  soever  we 
conceive  unity  to  be  divided,  no  fraction  can  be  assigned  suffi- 
ciently small  to  measure  at  the  same  time  this  root  and  unity. 
The  roots  of  numbers,  which  are  not  perfect  squares,  are  on  this 
account  called  incommensurable  or  irrational  quantities.  They 
are  sometimes  also  called  surds. 

To  indicate  that  the  square  root  of  a  quantity  is  to  be  taken, 
we  use  the  character  /^,  which  is  called  a  radical  sign.  Thus 
y\/l6  is  equivalent  to  4.  /k^2  is  an  iiicommensurable  or  surd 
quantity. 

EXTRACTION  OF  THE  SQUARE  ROOT  OF  FRACTIONS. 

96.  Since  a  fraction  is  raised  to  the  second  power  by  rais- 
ing the  numerator  to  the  second  power,  and  the  denominator 
to  the  second  power,  it  follows  that  the  square  root  of  a  frac- 
tion will  be  found  by  extracting  the  square  root  of  the  numera- 

9       3 
tor,  and  of  the  denominator.     Thus,  the  square  root  of  -r-r  is  -. 


EXTRACTION  OF  THE  SQUARE  ROOT.  125 

If  either   the   numerator   or   denominator  of  the   fraction  is 

not  a  perfect  square,  the  root  of  the  fraction  cannot  be  found 

exactly.     We  may,  however,  always   render   the   denominator 

of  a  fraction  a  perfect   square   by  multiplying  both  terms  of 

the    fraction    by   the    denominator.      This   will    not   alter   the 

/    value  of  the  fraction.     The  root  of  the  denominator  may  then 

be   found,   and    for   that  of  the  numerator,  we  must   take   the 

number   nearest  the  root.     Thus,  if  it  be  required  to  extract 
3 

the  square  root  of  -,  multiplying  both  terms  by  5,  the  fraction 
o 

becomes  7^,  the   root  of  which  is  nearest  -,  accurate  to  within 

less  than  -. 
o 

If  the  denominator  of  the  fraction  contain  a  factor,  which 
is  a  perfect  square,  it  will  be  sufficient  to  multiply  both  terms  by 
the  other  factor  of  the  denominator.     Thus,  let.  it  be  required 

Q 

to  find  the  square  root  of  — ;  multiplying  both  terms  by  6,  the 

48  .     7 

fraction  becomes  k— 7,  the  root  of  which  is  —^,  accurate  to  within 

oJ4  Jo 

less  than  —^. 
10 

If  a**  greater  degree  of  accuracy  is  required,  we  convert  the 
fraction  into  another,  the  denominator  of  which  is  a  perfect 
square,  but  greater  than  that  obtained  by  the  method  above. 

3  1 

To  find,  for  example,  the  square  root  of  -  to  within  -r^,  the 

o  15 

fraction  must  be  converted  into  225ths.     This  is  done  by  multi- 

3       135 

plying  both  terms  by  45.     Thus  we  have  -  =  — — ,  the   root   of 

5       225 

which  is  nearest  v?»  accurate  to  withm  less  than  — -. 
15  15 

After  making  the  denominator  a  perfect  square,  we  may  mul- 
tiply both  terms  of  the  proposed  fraction  by  any  number, 
which  is  a  perfect  square,  and  thus  approximate  the  root 
more   nearly.      If,    for  example,  we  multiply   both  terms  of 


126  ELEMENTS    OF    ALGEBRA. 

-^  by  144,  the  square  of  12,  we  obtain  75^7^,  the  root  of  which 

4-fi  *^ 

is  nearest  7^7:.      Thus,  we   have  the  root  of  -   to  within   less 
60  0 

thanl. 

97.  We  may  in  this  way  approximate  the  roots  of  whole 
numbers,  the  roots  of  which  cannot  be  exactly  assigned. 

If  it  be  required,  for  example,  to  find  the  square  root  of  2 ; 
we  convert  it  into  a  fraction  the  denominator  of  which  will  be 

a  perfect  square.     Thus,  if  we  put  2  =  ^— -,  we  have  for  the 

root  —  or  1^^,  accurate  to  within  less  than  —=. 
lo  10 

In  general,  to  find  the  square  root  of  a  number  accurate  to 
within  a  given  fraction,  we  multiply  the  proposed  number  by  the 
square  of  the  denominator  of  the  giveii  fraction  ;  we  then  fina 
the  entire  part  of  the  square  root  of  this  product,  and  divide  the 
result  by  the  denominator  of  the  given  fraction. 

This  rule  may  be  demonstrated  as  follows.  Let  a  be  the  num- 
ber proposed,  and  let  it  be  required  to  find  the  root  of  a  to  within 

less  than!. 
n 

a  7l 
We  shall  have,  it  is  evident,  a  =  —^ ;  let  r  be  the  entire  part 

of  the  root  of  the  numerator  of  an^;  an^  will  be  comprised  between 
r^  and  (r  -f-  1 Y,  and  by  consequence  the   square  root  of  a  will 

be  comprised  between  those  of  — ^  and  - — -^ — ,  that  is  to  say, 

r         r   I    1  7* 

between  -  and       "^     ;  whence  -  will  be  the  root  of  a  to  within 
n  n  n 

less  than  l 
n 

98.  To  approximate  the  root  of  a  number,  which  is  not  a 
perfect  square,  it  will  be  most  convenient  to  employ  some  power 
oT  10  as  the  multiplier  of  the  proposed,  or  which  is  the  same 
thing,  to  convert  the  proposed  into  a  fraction,  the  denominator  of 


EXTRACTION  OF  THE  SQUARE  ROOT.  127 

which   shall    be   some   power  of    10.      Thus,  to    approximate 
the  root  of  2,  let  us  put  2  =  -r^  or  2.00,  the  approximate  root 

will  be  1.4.     Again,  let  2  =   ^      or  2.0000,  the  approximate 

root  will  be  1.41. 

99.  From  w^hat  has  been  done,  and  indeed  from  tl^e  nature 
of  multiplication  it  follows,  that  the  number  of  decimal  places 
in  the  power  will  be  double  the  decimal  places  in  the  root.  To 
find  the  approximate  root  of  an  entire  number  by  the  aid  of 
decimals  therefore,  we  must  annex  to  this  number  twice  as  many- 
zeros  as  there  are  decimal  places  Avanted  in  the  root.  Thus, 
if  5  places  are  required  in  the  root,  ten  zeros  must  be  annexed. 
The  zeros  may  be  annexed  as  we  proceed,  it  being  observed, 
that  two  zeros  must  be  annexed  for  every  new  figure  placed  in 
the  root. 

The  root  of  7,  to  three  places,  will  be  found  as  follows. 
7  (2.645 
4 


300 
276 


2400 
2096 

30400 
26425 

3975 

If  the  proposed  be  already  a  decimal,  the  number,  of  decimal 
places  must  be  made  even  by  annexing  a  zero,  if  necessary.  If 
the  root  of  the  number,  thus  prepared,  is  not  sufficiently  exact, 
two  zeros  must  be  annexed  for  every  new  figure  required  in  \he 
root. 

100.  To  find  the  root  of  a  vulgar  fraction  by  the  aid  of  deci 


128  ELEMENTS    OF   ALGEBRA. 

mals,  we  convert  this  fraction  into  a  decimal  and  then  extract  the 
root. 

If  the  proposed  consist  of  an  entire  part  and  a  fraction,  we 
convert  the  fraction  into  a  decimal,  annex  it  to  the  entire  part, 
and  then  extract  the  root. 

In  converting  the  fraction  into  a  decimal,  it  will  be  necessary 
to  pursue  the  operation,  until  twice  as  many  decimals  are  obtained, 
as  are  wanted  in  the  root. 

EXAMPLES. 

1.  Find  the  square  root  of  11  to  within  less  than  — 

It 

Ans.  3 

2.  Find  the  square  root  of  223  to  within  less  than  —- 


15' 

15' 


40* 
Ans.  14^. 

3.  Find  the  square  root  of  7  to  within  .01.  Ans.  2.64. 

4.  Find  the  square  root  of  227  to  within  .0001. 

Ans.  15.0665. 

5.  Find  the  square  root  of  j^  to  3  places  of  decimals. 

Ans.  0.645. 
3 

6.  Finu  the  approximate  square  root  of  1~.     Ans.  1.32 -[-• 

13 

7.  Find  the  approximate  square  root  of  2  —=. 

Ans.  1.6931 +. 

8.  Find  the  approximate  square  root  of  31.027. 

Ans.  5.57+. 

9.  Find  the  approximate  square  root  of  0.01001. 

Ans.  0.10004 +. 

10.  Find  the  approximate  square  root  of  3271.4707. 

Ans.  57.19+. 


SQUARE    ROOT   OF   ALGEBRAIC   QUANTITIES.  129 


EXTRACTION   OF   THE    SQUARE    ROOT   OF    ALGEBRAIC    QUANTITIES. 

101.  By  the  rule  for  multiplication  we  have 

A  monomial  is  therefore  raised  to  the  square  by  squaring  the 
coefficient  and  doubling  the  exponent  of  each  of  the  letters. 
Whence  to  extract  the  square  root  of  a  monomial,  it  is  necessary 
1°.  to  extract  the  root  of  the  coefficient ;  2°.  to  divide  the  expo- 
nents of  each  of  the  letters  by  2. 

According  to  this  rule,  we  have 

V6^7¥=San\  * 

^/625^¥?  =  25abU\ 

In  order  that  a  monomial  may  be  a  perfect  square,  its  coeffi- 
cient, it  is  evident  from  the  preceding  rule,  must  be  a  perfect 
square  and  the  exponent  of  each  of  the  letters  must  be  an  even 
number. 

Thus  98a 5*  is  not  a  perfect  square.  Its  root  can,  therefore, 
be  only  indicated  by  means  of  the  radical  sig-n,  thus  ^98  a  b*. 
Expressions  of  this  kind  are  called  irrational  quantities  of  the 
second  degree,  or  more  simply  radicals  of  the  second  degree. 

102.  The  second  power  of  a  product,  it  is  easy  to  see,  is  the 
same  as  the  product  of  the  second  powers  of  all  its  factors.  It 
follows,  therefore,  that  the  square  root  of  a  product  will  be  the 
same  as  the  product  of  the  square  root  of  all  its  factors. 

By  means  of  this  principle,  we  may  frequently  reduce  to  a 
more  simple  form  expressions  of  the  kind,  which  we  are  here 
considering.  Thus,  the  above  expression  w98ab*  may  be  put 
under  the  form  \/Wb*X^^;  but  ^/49^=7Z>^  whence 
^/98^'=7bW2^. 

In  like  manner,  we  have 
V864a'^'c"  =  Vl44a^i*c'»X6^c=  12aiV  V6bl 


130  ELEMENTS    OF   ALGEBRA. 

In  the  expression  1b^  w2aj  \2ab^c^ ^/Qbc,  the  quantities  73', 
\2ab'^c^  placed  without  the  radical  sign  are  called  the  coefficients 
of  the  radical.  The  expressions  themselves  are  said  to  be 
reduced  to  their  most  simple  form. 

From  what  has  been  done,  we  have  the  following  rule  for 
reducing  irrational  quantities,  consisting  of  one  term,  to  their 
most  simple  form,  viz.  Separate  the  quantity  proposed  into  two 
parts,  one  of  lohich  shall  contain  all  the  factors,  which  are  perfect 
squares,  and  the  other  those  lohich  are  not ;  write  the  roots  of  the 
factors,  which  are  perfect  squares,  without  the  radical  sign  as 
multipliers  of  the  radical  quantity,  and  retain  under  the  radical 
sign  the  factors,  which  are  not  perfect  squares. 

1.  To  reduce  ^/ida^bc  to  its  most  simple  form. 

Ans.  5a^3abc. 

2.  To  reduce  ^S2a^b^c  to  its  most  simple  form. 

Ans.  4.aHW2c'. 

3.  To  reduce  ^  175  a^b^c''d  to  its  most  simple  form. 

Ans.  5aHcWTbcd, 

4.  To  reduce  ^^Oda^b^c^de  to  its  most  simple  form. 

Ans.   9ab^c\^5ade, 

5.  To  reduce  \f  29^.c^ b'^ & d^ ^  to  its  most  simple  form. 

Ans.  1  d^b^cde'^^abce. 

6.  To  reduce  VsiTo^^V^ to  its  most  simple  form. 

Ans.  Wd^b^edfsfr^lxd, 


7.  To  reduce  's/Vd\^:(£'b^(^d  to  its  most  simple  form. 

Ans.  Via^b'^c's/^abcd. 

103.  The  square  of  —  a,  it  will  be  observed,  is  c^,  as  well 
as  that  of  A- a;  the  root  therefore  of  c?,  may  be  either  -^-aon 
—  a.  Both  of  these  roots  may  be  comprehended  in  one  expres- 
sion by  means  of  the  double  sign  ±.     Thus 


SQUARE  ROOT  OF  ALGEBRAIC  QUANTITIES.        131' 

The  double  sign,  it  is  evident,  should  be  considered  as  ajSect- 
ing  the  square  root  of  all  quantities  whatever. 

If,  the  monomial  proposed  be  negative,  the  square  root  i»' 
impossible ;  since  there  is  no  quantity,  positive  or  negative, 
which  multiplied  by  itself  will  produce  a  negative  quantity. 
Thus,  a/  —  a, /v/  —  2b^  are  impossible  or  imaginary  quanti- 
ties. 

Expressions  of  this  kind  may  be  simplified  in  the  same  man- 
ner as  radical  expressions,  which  are  real.  Thus  V  —  9  may 
be  put  under  the  form  \^  —  1  X  9 ;  whence 

In  like  manner  \/  —  4<z^  =  2flV  —  1. 

104.  We  proceed  to  the  extraction  of  the  square  root  of 
polynomials. 

A  quantity  consisting  of  two  terms  cannot,  it  is  evident,  be  a 
perfect  square,  for  the  square  of  a  simple  quantity  will  be  a  sim- 
ple quantity,  and  the  square  of  a  binomial  consists  always  of 
three  terms. 

This  being  premised,  let  the  proposed  be  a  trmomial,  its  root, 
it  is  evident,  will  consist  of  at  least  two  terms.  Let  w  -|-  w  be 
the  root,  we  have  {m  -\-  rif  =  7r?  -\-  2m7i  -J-  n?. 

This  shows,  that  if  the  proposed  be  arranged  with  reference  to 
the  powers  of  some  letter  that,  P.  the  first  term  of  the  proposed 
will  be  the  square  of  the  first  term  of  the  root  sought;  2°.  the 
second  term  of  the  proposed  will  be  equal  to  twice  the  first 
term  of  the  root  multiplied  by  the  second;  3°.  the  third  term 
of  the  proposed  will  be  the  square  of  the  second  term  of  the 
root. 

Let  it  be  proposed  to  extract  the  root  of  the  trinomial 
2^a^b^c+\Qa^(?-\-^b\ 

Arranging  with  reference  to  the  letter  a,  the  proposed  be^ 
comes  16a*c=  +  2^aH^c  +  9^»^ 

In  order  to  obtain  the  root,  we  extract  according  to  what 
has  been  said  the  root  of  the  first  term  16  aV,  which  gives 


132  ELEMENTS   OF    ALGEBBA. 

4fl'c.  This  is  the  first  term  of  the  root.  Dividing  next  the 
second  term  2^(^b^c  by  Sc^c,  twice  the  term  of  the  root  already 
found,  we  have  3^'  for  the  second  term  of  the  root,  and  since 
the  square  of  this  is  equal  to  93^  the  remaining  term  of  the 
proposed,  the  proposed  is  a  perfect  square,  the  root  of  which 
IS  ^0^0  + W.    ^  f 

Again,  let  the  proposed  consist  of  more  than  three  terms, 
its  root  will  consist  of  more  than  two  terms.  Let  it  consist 
of  three  and  let  m-\-n-\-p  be  the  root.  The  expression 
m-\-n-\-p  may  be  put  under  the  form  {m •\- n)  -\- p ;  forming 
the  square  after  the  manner  of  a  binomial,  we  have  for  the  re- 
sult {m  -f-  Tif  -\-  2  [m  -\-  n)  p  -\-  p^,  or  developing  [m  -\-  rif  the 
result  will  be  tt^  -\-  ''Zmn  -\~  rl^  ~\-  2  {m  '\-  n)  p  -^  p^.  The  pro- 
posed, therefore,  being  arranged  with  reference  to  the  powers 
of  some  letter,  it  is  evident,  that  the  first  term  of  the  root  will 
be  found  by  extracting  the  root  of  the  first  term  of  the  proposed, 
and  that  the  second  term  of  the  root  will  be  found  by  dividing 
the  second  term  of  the  proposed  by  twice  the  first  term  of  the 
root  already  found.  If,  then,  we  subtract  from  the  proposed 
the  square  of  the  two  terms  of  the  root  already  obtained,  the 
remainder  will  be  equal  to  twice  the  first  two  terms  of  the 
root  multiplied  by  the  third  plus  the  square  of  the  third.  Divid- 
ing this  remainder,  therefore,  by  twice  the  terms  of  the  root 
already  found,  or  which  is  the  same  thing,  dividing  the  first 
term  of  the  remainder  by  twice  the  first  term  of  the  root,  we 
shall  obtain  the  third  term  sought.  Subtracting  from  the  first 
remainder  twice  the  product  of  the  first  two  terms  of  the  root 
by  the  third,  together  with  the  square  of  the  third,  if  the  result 
be  0,  the  proposed  is  a  perfect  square,  and  the  root  is  exactly 
obtained. 

Let  it  be  proposed  to  find  the  square  root  of  the  polynomial 

The  proposed  being  arranged  with  reference  to  the  letter  0, 
the  work  will  be  as  follows : 


SQUARE  ROOT  OF  ALGEBRAIC  QUANTITIES.        133 

25a*^20aH4-4:9aH^  —  24:ab'+16b*)5^-^^^nb+A^ 

A0a'b^  —  24:ab^  +  l6b* 
maH^  —  24:ab^  +  l6b* 

0 

We  begin  by  extracting  the  root  of  25  a*,  this  gives  5a^  for 
the  first  term  of  the  root  sought,  which  we  place  at  the  right 
of  the  proposed  and  on  the  same  line  with  it ;  we  then  multiply 
this  term  of  the  root  by  2  and  write  the  result  10  a^  under  the 
root.  Dividing  next  the  second  term  of  the  proposed  by  10  a^  we 
obtain  — Sab  for  the  second  term  of  the  root  sought.  Squaring 
the  part  of  the  root  already  found,  viz.  5a^^-3ab,  and  subtract- 
ing the  square  from  the  proposed,  we  have  for  the  first  term  of 
the  remainder  4.0  a^  P.  Dividing  this  last  by  10  a*  the  double  of 
5 a',. we  obtain  4^^  for  the  quotient. 

This  is  the  third  term  of  the  root  sought;  forming  next  the 
double  product  of  5a*  —  2ab  by  45*,  and  subtracting  the  result 
together  with  the  square  of  45*  from  the  first  remainder  the  re- 
sult is  0.  The  proposed  is,  therefore,  a  perfect  square,  and  we 
have  for  the  root  required 

5a*  — 3a5  +  45*. 

The  calculations  in  the  above  example  may  be  performed  with 
more  facility  as  follows. 


25  a'- 
25  a* 

-30a^5  +  49a*5*  — 24a53+165* 

5a*  — 3a5  +  45» 
lOa^  — 3a5 

" 

-30a35  +  49a*5* 
-SOaH+    9a*  5* 

10^2  — 6a5  +  45* 

40a*5*  — 24a53+165* 
40a*5*  — 24a53-j-165* 

0 

Having  found  the  first  term  5c^  of  the  rooot,  we  subtract  its 
square  from  the  first  term  of  the  proposed,  and  bring  down  the 
next  two  terms  for  a  dividend.     Dividing  the  first  term  of  th« 

L 


434  ELEMENTS   OF  ALGEBRA. 

dividend  by  10 a^,  we  obtain  —  2ab,  the  second  term  of  the  root; 
this  we  place  by  the  side  of  10  a^;  we  then  muhiply  the  whole, 
viz.  10 a^ —  3a^  by  this  second  term  and  subtract  the  result  from 
the  dividend,  which  gives  a  remainder  40a^b^;  to  this  remainder 
we  bring  down  the  two  remaining  terms  of  the  proposed  for  a 
new  dividend.  Doubling  the  two  terms  of  the  root  already 
'found  for  a  new  divisor,  we  write  the  result  under  10  a^;  divid- 
'ing  next  the  first  term  of  the  new  dividend  by  the  first  term  of 
the  divisor,  we  obtain  4:b^  the  third  term  of  the  root,  which  we 
place  by  the  side  of  the  last  divisor ;  we  then  multiply  the  whole 
by  this  last  term  of  the  root,  and  subtracting  the  result  from  the 
ilast  dividend,  0  remains. 

105.  The  same  process,  it  is  easy  to  see,  may  be  extended  to 
a  polynomial  of  any  number  of  terms  whatever. 

EXAMPLES. 

1.  To  find  the  square  toot  of 

4a^  +  12ffl3 :»  +  13aV  +  6fla:3 -f- a^. 

Ans.  2a^-\-2aX'\- a?, 

2.  To  find  the  square  root  of 

9x*—12x^  +  16x^  —  Sx  +  4.. 

Ans.  Saf*-— 2a;  +  2. 

3.  To  find  the  square  root  of 

4x'  ^16x^-{-  24ar^  —  16a:  +  4. 

Ans.  2a^  —  4:X  +  2. 

4.  To  find  the  square  root  of 

a;«  _[_  4  a:5  +  lOz^  +  SOa:^  +  25r^  +  24:r  +  16. 

Ans.  a;3  +  2ar^  +  3a;  +  4. 
6.  To  find  the  square  root  of 

4a:'' +  12a:5  +  5a:*  —  2a;^  +  7ar^  —  2a:  +  1. 

Ans.  2a,-^  +  3ar^  — a:+l. 

106.  The   polynomial   proposed  being  arranged  with   refer- 
ence to  the  powers   of  some  letter,  if  the  first  term  of  the 


SQUARE    ROOT    OF    ALGEBRAIC    QUANTITIES.  135 

proposed  is  not  a  perfect  square,  or  if  in  the  course  of  the 
operation  we  arrive  at  a  remainder,  the  first  term  of  which  is 
not  divisible  by  twice  the  first  term  of  the  root,  the  proposed 
is  not  a  perfect  square,  and  the  root  cannot  be  exactly  as- 
signed. 

The  polynomial  a^b-{- 4:0,^5"^ -\- 4:ab^,  for  example,  is  not,  it 
is  easy  to  see,  a  perfect  square ;  the  root  therefore  can  only 
be  indicated  thus,  ^/a■^^ -\- 4.d^b'^ -\- 4^ab'^.  We  may,  however, 
apply  to  expressions  of  this  kind  the  same  simplifications,  that 
have  already  been  applied  to  monomials.  The  proposed  in- 
deed may  be  put  under  the  form  w{d^-\-4:ab-\-4:b^)ab; 
but  the  root  of  a^-\-  Aab -\- 4tb^  is  evidently  a-\-2b,  whence 

^/a?b-\-4aH^-{-^ab^  z={a-\-2b)  s/'ab. 

EXAMPLES. 

1.  To  find  the  square  root  of  3  a*  &  —  6  a^  i'^  +  3  a'^  i^     . 

Ans.  a{a^b)^W. 

2.  To  find  the  square  root  oi5a^b  —  mab^  +  46 b\ 

Ans.  (a  — 3Z>)\/5^ 

3.  To  find  the  square  root  of  12 aV/  +  \2aH^  -\-2ab\ 

Ans.  b[2a-\-b)^^. 

4.  To  find  the  square  root  o^  a?  ~\-^aH  ^^ah" -[-b\ 

Ans.  [a  +  b)  Va  +  b. 

5.  To  find  the  square  root  of  a^  -{-a^b  —  ab"^  —  P. 

Ans.  [a  -)-  b)  w  a  —  b. 


SECTION   XII. — EQUATIONS  of  the  Second  Degree. 

107.  An  equation  is  said  to  be  of  the  second  degree,  when  it 
contains  the  second  power  of  the  unknown  quantity,  without  any 
of  the  higher  powers. 


136  ELEMENTS    OF   ALGEBRA. 

In  an  equation  of  the  second  degree  there  can  be,  therefore, 
three  kinds  of  terms  only,  viz.  1".  terms,  which  involve  the 
second  power  of  the  unknown  quantity,  2°.  terms,  which  in- 
volve the  first  power  of  the  unknown  quantity,  3°.  terms  con- 
sisting entirely  of  known  quantities. 

An  equation,  which  contains  all  three  of  these  different  kinds 
of  terms  is  called  a  complete  equation  of  the  second  degree. 

If  the  second  of  these  different  kinds  of  terms  be  wanting,  the 
equation  is  then  called  an  incoviplete  equation  of  the  second 
degree. 

A  complete  equation  of  the  second  degree  is  sometimes  called 
an  affected  equation,  and  an  incomplete  equation  is  sometimes 
called  a  pure  equation  of  the  second  degree. 

108.  We  are  now  prepared  for  the  solution  of  incomplete 
equations  of  the  second  degree. 

Let  there  be  proposed,  for  example,  the  equation 

3r^_29  =  ^  +  510. 

Freeing  from  denominators,  we  have 

12ar^— 116  =  ar^  + 2040; 
transposing  and  uniting  terms 

llar^  =  2156, 
or  r^=196, 

whence,  extracting  the  foot  of  both  members 

:i=:14. 
Equations  of  the  second  degree,  it  should  be  observed,  admit 
of  two  values  for  the  unknown  quantity,  while  those  of  the  first 
degree  admit  of  but  one  only.  This  arises  from  the  circum- 
stance, that  the  second  power  of  a  quantity  will  be  positive, 
whether  the  quantity  itself  be  positive  or  negative. 

Thus  we  have  x  in  the  preceding  example  equal  -f"  14  or 
—  14,  or,  uniting  both  values  in  one  expression,  we  have 
2:  =  ±14. 


EQUATIONS  OF   THE   SECOND  DEGREE.  137 

Let  US  talce,  as  a  second  example,  the  equation 

Freeing  from  denominators,   transposing  and  reducing,   we 
have 


ar  =  -^,  whence  x 


Y       29' 


252 

In  this  example  -^  is  not  a  perfect  square ;  we  can  therefore 

obtain  only  an  approximate  value  for  x. 

Let  us  take,  as  a  third  example,  the  equation 
ar^+25  =  9. 

Deducing  the  value  of  x  from  this  equation,  we  have 
•  x=^^f'^^^^. 

To  find  the  value  of  ar,  we  are  here  required  to  extract  the 
square  root  of  —  16.  But  this  is  impossible ;  for,  as  there  is  no 
quantity  positive  or  negative,  which  multiplied  by  itself  will 
produce  a  negative  quantity,  -^  16,  it  is  evident,  cannot  have 
a  square  root  either  exact  or  approximate.  —  16  may  indeed  be 
considered  as  arising  from  the  multiplication  of  -f-4  by  — 4; 
but  -\-  4  and  —  4  are  different  quantities ;  their  product  therefore 
is  not  a  square. 

The  result  a:  =  \/ — 16  shows  then,-  that  it  is  impossible  to 
resolve  the  equation,  from  which  it  is  derived.  In  general,  an 
expression  for  the  square  root  of  a  negative  quantity  is  to  be 
regarded  as  a  symbol  of  impossibility. 

109.  Equations  of  the  kind,  which  we  are  here  considering, 
may  always  be  reduced  to  an  equation  of  the  form  ax'^  =  b^  a  and 
h  denoting  any  known  quantities  whatever,  positive  or  negative. 
It  is  evident,  that  they  may  be  reduced  to  this  state,  by  collecting 
into  one  member  the  tervis,  ivhich  involve  y^  and  reducing  them 
to  one  term,  and  collecting  the  known  terms  into  the  other 
member. 


13S  ELEMENTS   OF   ALGEBRA. 


v/ 


Resolving  the  equation  ax^  =  b^  we  have 

T 
a 

This  is  a  general  solution  for  incomplete  equations  of  the 
second  degree. 

If  -  be  a  perfect  square,  the  value  of  x  may  be  obtained  ex- 
a 

actly,  if  not,  it  may  be  found  with  such  degree  of  approximation 
as  we  please.  If  -  be  negative,  we  shall  have!  y^ a  sym- 
bol of  impossibility. 

From  what  has  been  done,  we  have  the  following  rule  for  the 
solution  of  incomplete  equations  of  the  second  degree,  viz.  Col- 
lect into  one  member  all  the  terms,  which  involve  the  square  of  the 
unknown  quantity,  and  the  known  quantities  into  the  other 4  free 
the  square  of  the  unknown  quantity  from  the  quantities,  by  which 
it  is  multiplied  or  divided ;  the  value  of  the  unknown  quantity 
will  then  be  obtained  by  extracting  the  square  root  of  each  memler. 

QUESTIONS  PRODUCING  INCOMPLETE  EQUATIONS  OF  THE 
SECOND  DEGREE. 

1.  What  two  numbers  are  those,  whose  difference  is  to  the 
greater  as  2  to  9,  and  the  difference  of  whose  squares  is  128  ? 

Let  9  a:  =  the  greater  and  2  a:  =  the  difference,  then,  &c. 

Ans.  18  and  14. 

2.  It  is  required  to  divide  the  number  14  into  two  such  parts, 
that  the  quotient  of  the  greater  part  divided  by  the  less  may  be 
to  the  quotient  of  the  less  divided  by  the  greater  as  48  to  27. 

Let  X  =  the  greater,  then  14  —  x=  the  less,  and  we  have 

14  —  X  X 

or  272r^  =  48(14  — a:)'^;      , 

dividing  by  3  to  make  the  coefficients  perfect  squares 

9a:'=16(14  — a:)=^; 
whence  8a:  =  4  (14  —  x).  Ans.  8  and  6. 


EQUATIONS   OF   THE    SECOND   DEGREE.  100 

3.  It  is  required  to  divide  the  number  18  into  two  such  parts, 
4hat  the  squares  of  these  parts  may  be  in  the  proportion  of  25  to 
16.  Ans.  10  and  8. 

4.  In  a  court  there  are  two  square  grass  plots ;  a  side  of  one 
of  which  is  10  yards  longer  than  the  side  of  the  other ;  and  their 
areas  are  as  25  to  9.     What  are  the  lengths  of  the  sides  ? 

Ans.  25  and  15  yards. 

5.  A  person  bought  two  pieces  of  linen,  which  together  mea- 
sured 36  yards.  Each  of  them  cost  as  many  shillings  a  yard  as 
there  were  yards  in  the  piece ;  and  their  whole  prices  were  in 
the  proportion  of  4  to  1.     What  were  the  lengths  of  the  pieces  ? 

Ans.  24  and  12  yards. 

6.  There  is  a  rectangular  field,  whose  length  is  to  the  breadth 
in  the  proportion  of  6  to  5.  A  part  of  this  equal  to  |-  of  the 
whole  being  planted,  there  remain  for  ploughing  625  square 
yards.     What  are  the  dimensions  of  the  field  ? 

Ans.  The  sides  are  30  and  25  yards. 

7.  Two  workmen,  A  and  B,  were  engaged  to  work  for  a 
certain  number  of  days  at  different  rates.  At  the  end  of  the 
time,  A  who  had  played  4  of  the  days,  received  75  shillings,  but 
B  who  had  played  7  of  the  days,  received  only  48  shillings. 
Now  had  B  played  4  days,  and  A  played  7  days,  they  would 
have  received  exactly  alike.  For  how  many  days  were  they 
engaged ;  how  many  did  each  work,  and  what  had  each  per  day  ? 

Ans.  19  days ;  A  worked  15,  and  B  12  dajj's,  and 
A  received  5s.  and  B  4s.  a  day. 

8.  Two  travellers,  A  and  B,  set  out  to  meet  each  other,  A 
leaving  the  town  C  at  the  same  time  that  B  left  D.  They 
travelled  the  direct  road  C  D,  and  on  meeting,  it  appeared  that 
A  had  travelled  IS  miles  more  than  B ;  and  that  A  could  have 
gone  B's  journey  in  15J  days,  but  B  would  have  been  28  days 
in  performing  A's  journey.  What  was  the  distance  between  C 
and  D  ?  Ans.  126  miles. 

9.  A  and  B  carried  100  eggs  between  them  to  market  and 
each  received  the  same  sum.     If  A  had  carried  as  many  as  B 


140  ELEMENTS    OF   ALGEBRA. 

he  would  have  received  18  pence  for  them,  and  if  B  had  carried 
only  as  many  as  A,  he  would  have  received  only  8  pence.  How 
many  had  each  ?  Ans.  A  40,  B  60. 

10.  What  two  numbers  are  those,  whose  sum  is  to  the  greater 
as  11  to  7,  the  difference  of  their  squares  being  132? 

Ans.  8  and  14. 

11.  A  merchant  sold  for  $960  a  certain  number  of  pieces  of 
silk,  for  which  he  paid  four-fifths  as  many  dollars  a  piece  as  the're 
were  pieces.  He  gained  $1000  by  the  sale,  how  many  pieces 
did  he  sell  ?  Ans.  The  question  is  impossible. 

COMPLETE  EQUATIONS  OF  THE  SECOND  DEGREE. 

110.  Let  us  take  next  the  equation  a;'-}- 8a;  =  209.  This 
is  a  complete  equation  of  the  second  degree.  The  solution  of 
this  equation,  it  is  evident,  would  present  no  difficulty,  if  the  left 
hand  member  v/ere  a  perfect  square.  But  this  is  not  the  case ; 
for  the  square  of  a  quantity  -consisting  of  one  term  will  consist 
of  one  term,  and  the  square  of  a  quantity  consisting  of  two 
terms  will  contain  three  terms.  Let  us  then  see  if  a^-\-8x  can 
be  made  a  perfect  square ;  for  this  purpose,  it  will  be  recollected, 
that  the  three  parts  which  compose  the  square  of  a  binomial  are 
P.  the  square  of  the  first  term  of  the  binomial^  2°.  tioice  the  first 
term  multiplied  by  the  second,  3°.  the  square  of  the  second  term. 
Thus, 

[x-\-af  =  x'  +  2ax-\-a\ 

If,  then,  we  compare  a;*^  +  8a:  with  x^ -\-2ax-\- a^/\i  is  evi- 
dent that  af^-f-' 8  a:  may  Se  considered  the  first  and  second  terms 
in  the  square  of  a  binomial.  The  first  term  of  this  binomial 
will  evidently  be  x;  then  as  8  a;  must  contain  twice  the  first  term 
by  the  second,  the  second  will  be  found  by  dividing  8a;  by  2a:, 
which  gives  4  for  the  quotient,  ar^-f- 8a:  is,  therefore,  the  first 
two  terms  in  the  square  of  the  binomial  x-\-^.  If?  then,  we 
add  16,  the  square  of  4,  to  a;*^  +  8a:,  the  left  hand  member  of  the 
proposed,   the   result  ar^  +  8a:  +  16  will  be  a  perfect   square. 


EQUATIONS   OF   THE    SECOND    DEGREE.  141 

But  if  16  be  added  to  the  left  hand  member,  it  must  also  bo 
added  to  the  right  in  order  to  preserve  the  equality ;  the  proposed 
will  then  become 

a^-^Sx-^  16  =  225. 
Extracting  the  root  of  each  member  of  this  last,  we  have 
a:  +  4  =  ±15, 
whence  a;=ll,  a;  =  —  19. 

Let  us  take,  as  a  second  example,  the  equation 

2 

Comparing  a^  —  -^x  with  the  square  of  the  binomial  x  —  a, 
3 

2 

viz.  ar*  —  2ax-\-a^,itis  evident,  that  a^ — ^x  may  be  considered 

the  first  two  terms  of  the  square  of  a  binomial.  By  the  same 
course  of  reasoning  as  in  the  preceding  example,  we  find  this 

binomial  to  be  a:  —  ^r.      If,  then,  the  square  of  ^  be  added  to 
o  o 

both  sides,  the  left  hand  member  will  be  a  perfect  square,  and 
we  have 

Extracting  the  root  of  each  member,  we  have 
x-i  =  =fc4; 

whence  x=  4| ,  a:  =  —  3|. 

Let  us  take,  as  a  third  example,  the  equation  a^-\'px  =  q. 

Comparing   the   left   hand   member   of   this   equation   with 
a?-^-2aa:-}-a^  it  is  evident,  that  it  may  be  considered  as  the 

first  two  terms  in  the  square  of  the  binomial  a:  +  ^ ;  whence,  if 

the  square  of  ^  be  added  to  both  sides,  the  left  hand  member  will 
become  a  perfect  square,  and  we  shall  have 


142  ELEMENTS    OF   ALGEBEA. 

Extracting  the  root  of  each  member 

Making  the  left  hand  member  a  perfect  square  is  called  com- 
pleting  the  square.     This  is  done,  as  will  readily  be  inferred 
from  the  preceding  examples,  hy  adding  to  both  sides  the  square 
of  one  half  the  coefficient  of  x  in  the  second  term. 
Let  us  take  for  a  fourth  example,  the  equation 
3  61— ar^ 

5'^~'4x  — 2* 
Freeing  from  denominators,  we  have 

140  a;  —  70  —  12ar' +  6a;  =  305  —  5a?. 
Transposing  and  uniting  terms,  we  have 
146  a;  — Tar' =  375. 
Or,  changing  the  signs  of  each  term  and  dividing  by  the 
coefficient  of  a;^ 

146a;  _      375 
7  7  ' 

Completing  the  square,  we  have 

146a:      5329  _      375      5329  _  2704 
~7~  '     49   ~"        7     '     49   ~"  49  * 
Whence,  extracting  the  root  of  each  member 
__73__      52 

a;=17f,  a;  =  3. 

111.  The  rule  for  completing  the  square  applies  only,  it  is 
evident,  to  equations  of  the  form  :i^ -\-'px-=q,  p  and  q  denoting 
any  quantities  whatever,  positive  or  negative. 

If  not  already  of  the  form  x^  -\-px  =  q^  equations  of  the 
kind,  which  we  are  here  considering,  must  always  be  reduced 
to  this  form,  before  completing  the  square.  Thus,  in  the  pre- 
ceding example,  the  given  equation  was  reduced,  before  com* 


EQUATIONS   OF   THE    SECOND   DEGKEE. 


1^ 


,    .       ^                       /     146             375  .        ^  V 

pletmg  the  square,  to  ar —x  = =-,  an  equation  of  the 

form  required. 

It  is  evident,  that  all  complete  equations  of  the  second  degree 
may  be  reduced  to  the  form  a?  -\-pz-=-q,  1°.  by  collecting 
all  the  terms  which  involve  x  into  the  first  member  and  uniting 
the  terms,  which  contain  ar^,  into  one  term,  and  those  which 
contam  x  into  another,  2°.  by  changing  the  signs  of  each 
term,  if  necessary,  in  order  to  render  that  of  3?  positive,  3°.  by 
dividing  all  the  terms  by  the  multiplier  of  x^^  if  it  have  a  mul- 
tiplier, and  multiplying  all  the  terms  by  the  divisor  of  3?,  if  it 
have  a  divisor. 

CLX  C  X 

Let  the  equation  ~ 5ar^  =  — -j-ae  be  reduced  to  tho 

form  x^-\-px  =  q. 

Freeing  from  denominators,  we  have 

5ax-^20bx^  =  'icx-\'20ae 
By  transposition         — 20bx^ -\-5ax  —  4:cx==20ae 
Changing  signs  20bx^  —  5ax-\-4:Cx  =  —  20ae 

Uniting  terms  20b x^ — {5a  —  4c)  a:  =  —  20ae 

Dividing  by  20  A  :^  -  ^^^^  ^  =  "  y • 

Comparing  this  equation  with  the  general  formula,  we  have 

{5a  —  4c)  ae 

P= 203—'  «=-T- 

From  what  has  been  done,  we  have  the  following  rule  for 
the  solution  of  complete  equations  of  the  second  degree,  viz. 
]°.  The  equation  being  reduced  to  the  form  x'^-f-pxsrrq,  add 
to  both  members  the  square  of  half  the  coefficient  ofx  in  the  second 
term;  2°.  extract  the  square  root  of  both  members,  taking  care 
to  give  to  the  root  of  the  second  member  the  double  sign  ±  ; 
3°.  deduce  the  value  of  :s.from  the  equationt  which  arises  from 
the  last  operation. 


y^fk  ELEMENTS    OF    ALGEBBA. 


EXAMPLES. 


2a^  X 

1.  Given  -o-  +  ^s-  ==  «  +  8,  to  find  the  values  of  x. 

O  <6 


2.  Given  4a; =  46,  to  find  the  values  of  x. 


Ans.  a;  =  3,  or  —  2J. 
I  the  values  of  x. 
Ans.  X  =  12,  or  —  .75. 


40  27 

3.  Given ^r  -I =13,  to  find  the  values  of  x. 

X  —  5    '    X 


14 X 

4.  Given  4  a; r— r-  =  14,  to  find  the  values  of  x. 

x-f- 1 


Ans.  a;  =  9,  or  ly^. 
he  values  of  x. 
Ans.  a;  =  4,  or  —  If. 


X  7 

5,  Given  — t-ttr  =  7i ^j  to  find  the  values  of  x. 

a;-|-60       Sx  —  o 


Ans.  X  =  14,  or  —  10. 

,  =.'?JL. 

2        '    2a;  — 5 


6.  Given  — J^ \-  -^ ~  =  5|,  to  find  the  values  of  x, 


Ans.  x  =  5,  or  6.9. 

'•  a^d7+ t/lT,  +  11  i  '°  fi°d  'he  values  of  .  and  ,. 
Ans.  a;  =  —  46,  or  2 ;  y=15,  or  3. 

^*  ^ind^a^^'^-lt^V^I^  62  I  ^°  ^^^  ^^^  ^^^"^'  ^^^  ^"^  2/. 

Ans.  a;  =  5,  or  —  36^y  ;^y  =  3,  or  30^^^ . 

9.Give„?£±^^  =  2,_51±^l 
4a;  ^  10 


,  4a;  +  3y 
and  — _L_^  =  2/  — 2 


to  find  the  values  of 

a;  and  y. 

16 

Ans.  a;  =  5,  or  —  2^\ ;  y  =  4,  or  Ifff. 
112.     "We  pass  next  to  the  solution  of  some  questions. 
1.  To  find  a  number  such,  that  if  three  times  this  number  be 
added  to  twice  its  square,  the  sum  will  be  65. 


EQUATIONS   OF   THE    SECOND   DEGREE.  145 

Putting  X  for  the  number  sought,  we  have  by  the  question 

Dividing  by  2,  we  have  oi?  -\--x=--^. 

3  9       65       9 

Completing  the  square,  ^  +  2^  +  3;q  =  "2"^T6* 

T.  ,  .3  23 

bxtractmg  the  root  a:  -j-  ^  =  db  -j-, 

whence  a:  =  5,  a;  = ^. 

The  first  value  of  x  satisfies  the  question  in  the  sense,  in 
which  it  is  enunciated.  In  order  to  interpret  the  second,  it 
will  be  observed,  that  if  we  put  —  x  instead  of  x  in  the  equa- 
tion 23? -\'^x  =  Q5,  it  becomes   2r^  —  3 a;  =  65.     Resolving 

13 
this  equation,  we  obtain  a;  =  -^,  x  =  —  5,  values  of  a:,  which 

13 

differ  from  the  preceding  only  in  the  signs.     The  number  -^  will, 

tit 

therefore,  satisfy  the  conditions  of  the  question  modified  thus. 
To  find  a  number  such,  that  if  three  times  this  number  be 

subtracted  from  twice  its  square,  the  remainder  will  be  65. 

2.  A  person  bought  some  sheep  for  £  72 ;  and  found  if  he  had 

bought  6  more  for  the  same  money,  he  would  have  paid  £  1  less 

for  each.     How  many  did  he  buy  ? 
Let  X  =  the  number,  we  have 

Z?__2?_  — 1 

X       x-\-6 
from  which  we  obtain  x  =  18,  or  —  24.     To  interpret  the  nega- 
tive result,  we  write  —  x  for  x  in  the  equation,  which  becomes 
72  72 

__a;       —a:-f  6~    '      ' 
or  which  is  the  same  thins: 


•& 


72    _72_ 


X  —  6       X 

an  equation,  which  corresponds  to  the  following  enunciation 
10 


146  ELEMENTS    OF    ALGEBRA. 

A  person  bought  some  sheep  for  £72,  and  found  if  he  had 
bought  6  less  for  the  same  money,  he  would  have  paid  £  1  more 
for  each.     How  many  did  he  buy  ? 

The  negative  values  here  modify  the  proposed  questions,  in  a 
mariner  analogous  to  what  takes  place,  as  we  have  already  seen, 
in  equations  of  the  first  degree. 

3.  To  find  a  number  such,  that  if  15  be  added  to  its  square, 
the  sum  will  be  equal  to  eight  times  this  number. 

Putting  X  for  the  number  sought,  we  have  by  the  question 

a,-2-f  15==8a:. 
Resolving  this  equation,  we  have 

re  =  5,  2;  =  3.- 
In  this  example  both  values  of  x  are  positive,  and  ansfver 
directly  the  conditions  of  the  question,  in  the  sense  in  which  it 
is  enunciated. 

4.  To  find  a  number  such,  that  if  the  square  of  this  number 
be  augmented  by  5  times  the  number  and  also  by  6,  the  result 
will  be  2. 

Putting  X  for  the  number  sought,  we  have  by  the  question 

ar^4-5a;  +  6  =  2. 
Whence,  resolving  the  equation  we  have  r^ 

The  values  of  x  in  this  example  are  both  negative ;  the  ques- 
tion, therefore,  as  is  evident  from  inspection,  cannot  be  solved  in 
the  sense,  in  which  it  is  enunciated. 

If  instead  of  x  we  write  —  a;  in  the  equation  of  the  proposed 
it  becomes  z^  —  5a: -[-6  =  2,  from  which  we  obtain  a:=l, 
a:  =  4.  The  numbers  1  and  4  will,  therefore,  satisfy  the  con- 
ditions of  the  proposed  modified  thus, 

To  find  a  number  such,  that  if  five  times  this  number  be 
subtracted  from  its  square,  and  6  be  added  to  the  remainder, 
the  result  will  be  2. 

5.  To  divide  the  number  10  into  two  such  parts,  that  the 
product  of  these  parts  will  be  30. 


EQUATIONS   OF   THE    SECOND   DEGREE.  147 

Putting  X  for  one  of  the  parts,  10  —  x  will  be  the  other ;  w« 
have  therefore  by  the  question 

10a;  — a:' =30. 
Resolving  this  equation,  we  obtain 

a;  =  5 -f  V  —  5,  x  =  5-r-^^5. 

This  result  indicates,  that  there  is  some  absurdity  in  the  con- 
ditions of  the  question  proposed,  since  in  order  to  obtain  the 
value  of  X,  we  must  extract  the  root  of  a  negative  quantity,  which 
is  impossible. 

In  order  to  see  in  what  this  absurdity  consists,  let  us  exam- 
ine into  what  two  parts  a  given  number  should  be  divided,  in 
order  that  the  product  of  these  parts  may  be  the  greatest  pos- 
sible. 

Let  us  represent  the  given  number  by  p,  the  product  of  the 
two  parts  by  q,  and  the  difference  of  the  two  parts  by  d;  the 

greater  part  will  then  be  f  +  ^j   and  the  less  ^  —  -,  and  we 

shall  have 


(1+1)  (1-1)=^- 


p"      (f 

4       4^ 

Here  the  value  of  q,  it  is  evident,  will  be  greater  as  that  of 
d  is  less ;  the  value  of  q  will,  therefore,  be  the  greatest  possible 
when  d  is  zero,  that  is,  the  product  will  be  the  greatest  possible, 
when  the  difference  between  the  two  parts  is  zero,  or  in  other 
words,  lohen  the  two  parts  are  equal. 

The  greatest  possible  product,  which  can  be  obtained  by  di- 
viding 10  into  two  parts  and  taking  their  product  will  be  25. 
The  absurdity  of  the  question  above  consists,  therefore,  in  re- 
quiring, that  the  product  of  the  two  parts,  into  which  10  is  to  be 
divided,  should  be  greater  than  25. 

113.  The  following  questions  will  serve  as  an  exercise  for  the 
learner. 


148  ELEMENTS    OF   ALGEBRA. 

■  1.  There  is  a  field  in  the  form  of  a  rectang-ular  parallelogram, 
whose  length  exceeds  the  hreadth  by  16  yards,  and  it  contains 
960  square  yards.     Required  the  length  and  breadth. 

Ans.  40  and  24  yards. 

2.  There  are  two  numbers,  whose  difference  is  9,  and  their 
sum  multiplied  by  the  greater,  produces  266.  What  are  those 
mumbers  ?  Ans.  14  and  5. 

^.  A  regiment  of  soldiers,  consisting  of  1066  men,  is  formed 
into  two  squares,  one  of  which  has  four  men  more  in  a  side  than 
the  other.  What  number  of  men  are  in  a  side  of  each  of  the 
squares  ?  Ans.  21  and  25. 

4.  Two  partners,  A  and  B,  gained  £  18  by  trade.  A's  money 
was  in  trade  12  months,  and  he  received  for  his  principal  and 
gain  £26.  Also  B's  money,  which  was  £30,  was  in  trade 
16  months.     What  money  did  A  put  into  trade? 

Ans.  £20. 

5.  The  plate  of  a  looking  glass  is  18  inches  by  12,  and  is  to 
,be  framed  with  a  frame  of  equal  width,  whose  area  is  to  be  equal 
to  that  of  the  glass.     Required  the  width  of  the  frame. 

Ans.  Jpnches. 

6.  A  grazier  bought  as  many  sheep  as  cost  him  £60 ;  out  of 
which  he  reserved  15,  and  sold  the  remainder  for  £54,  gaining 
2  shillings  a  head  by  them.  How  many  sheep  did  he  buy,  and 
what  was  the  price  of  each  ?  ^• 

Ans.  75  sheep,  and  t^e  price  was  16^. 

7.  A  person  bought  two  pieces  of  clotK'  of  different  sorts; 
whereof  the  finer  cost  4  shillings  a  yard  more  than  the  other ; 
for  the  finer  he  paid  £18;  but  the  coarser,  which  exceeded 
the  finer  in  length  by  2  yards,  cost  only  £16.  How  many 
yards  were  there  in  each  piece,  and  what  was  the  price  of  a 
yard  of  each? 

Ans.  18  yards  of  the  finer,  and  20  of  the  coarser,  and 
the  prices  were  £  1  and  16s.  respectively. 

8.  Three  merchants,  A,  B,  and  C,  made  a  joint  stock,  by 


GENERAL   EQUATION   OF   THE    SECOND  DEGREE.  l49 

which  they  gained  a  sum  less  than  that  stock  by  £80 ;  A^s  share 
of  the  gain  was  £60,  and  his  contribution  to  the  stock  was  £17 
more  than  B's.  Also  B  and  C  contributed  together  £325.  How 
much  did  each  contribute  ? 

Ans.  75,  58,  and  267  pounds  respectively. 

9.  Two  messengers,  A  and  B,  were  dispatched  at  the  same 
time  to  a  place  90  miles  distant ;  the  former  of  whom  riding  one 
mile  an  hour  more  than  the  other,  arrived  at  the  end  of  his 
journey  an  hour  before  him.  At  what  rate  did  each  travel  per 
hour  ?  Ans.  A  10  miles,  B  9. 

10.  The  joint  stock  of  two  partners,  A  and  B,  was  $416.  A's 
money  was  in  trade  9  months  and  B's  sii»  months ;  on  dividing 
their  stock  and  gain,  A  received  $228,  and  B  $252.  What  was 
each  man's  stock  ?  Ans.  A's  $192,  B's  $224. 

11.  A  and  B  sold  130  ells  of  silk,  of  which  40  ells  were  A's 
and  90  B's,  for  $42.  Now  A  sold  for  a  dollar  J  of  an  ell  more 
than  B  did.     How  many  ells  did  each  sell  for  a  dollar  ? 

Ans.  B  sold  3  ells,  and  A  3J  for  a  dollar. 

12.  A  square  court-yard  has  a  rectangular  gravel  walk  round 
it.  The  side  of  the  court  wants  2  yards  of  being  6  times  the 
breadth  of  the  gravel-walk ;  and  the  number  of  square  yards  in 
the  walk  exceeds  the  number  of  yards  in  the  periphery  of  the 
court  by  164.     Required  the  area  of  the  court. 

Ans.  256  yards. 


SECTION  XIII. — Discussion  of  the  General  Equation  and 
OF  Problems  of  the   Second  Degree. 

114.  All  complete  equations  of  the  second  degree  may,  as 
we  have  already  seen,  be  reduced  to  an  equation  of  the  form 
ar^-|-  Pa;=;  Q,  P  and  Q  denoting  any  known  quantities  whatever, 
positive  or  negative.     Resolving  this  equation,  we  have 


150  ELEMENTS    OF   ALGEBRA. 

This  is  a  general  solution  for  equations  of  the  second  degree. 
We  shall  now  examine  the  circumstances,  which  result  from 
the  different  hypotheses,  which  may  be  made  upon  the  known 
quantities  P  and  Q.  This  is  the  object  of  the  discussion  of  the 
general  equation  of  the  second  degree. 

115.  Any  quantity,  which  substituted  for  the  unknown  quantity 
in  an  equation  of  the  second  degree  will  satisfy  it,  is  called  a  root 
of  the  equation. 

Before  proceeding  to  the  proposed  discussion,  we  shall  show, 
that  every  equation  of  the  second  degree  admits  of  two  values 
for  the  unknown  quantity,  or  in  other  words  of  two  roots,  and  of 
two  only.     In  order  lo  this  we  take  the  general  equation 

x'  +  Vx^Q,;       (1) 
Completing  the  square,  we  have 

p2  F         /         P\2  P 

p2 

Let  Q  -|-  — -  ==  M',  we  shall  then  have 

(=" + 1) = ^'' "'  G + 1)'  -  "^^  °- 

But  the  first  member  of  this  equation  being  the  difference 
between  two  squares,  it  may  be  put  under  the  form 

(:.  +  ?+ m)(:.  +  |-m):^0.        (2) 

This  fast  equation  is,  it  is  evident,  a  necessary  consequence  of 
equation  (1)  and  the  converse.  Each  of  these  equations  will  be 
satisfied  therefore  by  the  values  of  x,  which  satisfy  the  other, 
and  by  these  only.  But  since  the  left  hand  member  of  equation 
(2)  is  composed  of  two  factors,  this  member  will  become  zero 
if  either  of  its  factors  is  equal  to  zero,  and  thus  the  equation  will 
be  satisfied. 

P  P 

If  we  suppose  a; -|- n — M=:0,  we  shall  harve  x  =  —  ^--j-M. 

P  P 

If  we  suppose  a:  -f-  ^  +  M  =  0,  we  shall  have  z = — ^  —  IML 


GENERAL   EQUATION    OF   THE    SECOND   DEGREE.  151 

Or  substituting  for  M  its  value,  we  have 


Since  equation  (2)  can  be  satisfied  only  by  putting  for  x  a 
value  which  will  reduce  to  zero  one  or  the  other  of  the  two  fac- 
tors, of  which  the  left  hand  member  is  composed,  it  follows,  that 
every  equation  of  the  second  degree  admits  of  tivo  roots  or  values 
for  the  unknown  quantity  and  of  two  only. 

It  follows  also  from  what  has  been  done,  that  every  equation 
of  the  second  degree  may  be  decomposed  into  two  binomial  factors 
of  the  first  degree  with  respect  to  x,  having  a  for  a  common  term^ 
and  the  two  roots  taken  with  their  signs  changed,  for  the  second 
terms.  ^ 

Resolving  the  equation  3?  -\-Sx  —  209  =  0,  for  example,  we 
have  a:  =11,  x  =  — 19.  Either  of  these  values  will  satisfy 
the  equation.     We  have  also 

(a;— 11)  (2:  + 19)  =  2-='  + 8a:  — 209  =  0. 

If  we  add  together  the  two  general  values  for  a:,  found  above, 
the  sum,  it  is  evident,  will  be  — P;  if  we  multiply  them  together 
the  product  will  be  —  Q ;  whence  1°.  The  algebraic  sum  of 
the  two  roots  is  equal  to  the  coefficient  of  the  second  term  taken  with 
the  contrary  sign.  2°.  The  product  of  the  two  roots%  equal  to 
the  second  member  of  the  equation,  taken  also  ivith  a  contrary 
sign. 

116.  Let  us  new  proceed  to  Uie  discussion  proposed.  Re- 
suming the  value  of  a;,  obtained  from  the  general  equation 
ar* -j-  Pa;  =  Q,  we  have 

In  order  to  find  the  value  of  this  expression,  which  contains  a 
radical,  that  is  a  quantity  the  root  of  which  is  to  be  extracted, 
we  must  be  able  to  extract  the  root  either  exactly  or  by  approxi- 


152  ELEMENTS    OF    ALGEBRA. 

F  . 

mation ;  Q  -f-  -r-  the  quantity  placed  under  the  radical  sign  must, 

therefore,  be  positive.     But  -r-  will  necessarily  be  positive,  what- 

P 

ever  the  sign  of  P  may  be  ;  the  sign  of  the  quantity  Q,-\--t-  will, 

therefore,  depend  principally  upon  that  of  Q  or  the  quantity  in 
the  equation  altogether  known. 

1.  This  being  premised,  let  Q  in  the  first  place  be  positive. 
In  this  case  P  may  be  either  positive  or  negative,  and  the  general 
equation  may  be  written  under  the  two  forms  ^ 

a^-^Yx  =  +Q,  x'  —  'Px  =  +  Q, 
or  uniting  both  in  one 

x':i.Vx  =  +  Q; 
from  which  we  have 


I±v/q+?- 


p2 

Here  Q-\--t-  will  evidently  be  positive ;  the  value  of  x  may, 

therefore,  be  obtained,  either  exactly  or  with   such  degree  of 
approximation  as  we  please. 

With  respect  to  the  two  values  of  x,  the  first,  viz. 


==p|+\/q+? 


F  .      P 

will  be  positive,  for  the  square  root  of  -r-  alone  being  — ,  the  square 

P2  P 

root  of  Q  -f-  —  will  be  greater  than  — -,  the  value  of  x  will,  there- 

fore,  have  the  same  sign  with  the  radical  and  will  by  consequence 
be  positive.  This  value  will  answer  directly  the  conditions  of 
the  equation,  or  the  problem  of  which  the  equation  is  the  algebraic 
translation. 

P  /  F 

The  second  value  of  x,  viz.  x  =  ^  —  —  Xy^   ^"^"47'  ^^^^S 

also  necessarily  of  the  same  sign  with  the  radical,  will  be  essen- 


GENERAL   EQUATION   OF   THE    SECOND   DEGREE.  163 

tially  7iegative.     This   value,  though  it  satisfies   the   equation, 
will  not  answer  the  conditions  of  the  question,  from  which  the 
equation  is  derived.     It  belongs  to  an  analogous  question  corre 
spending  to  the  equation,  after  —  x  has  been  introduced  instead 
of  X,  that  is,  to  ar^  ^  Pa:=  Q.     Indeed,  from  this  last  equation 

P  /  W 

we  deduce  z  =  ±  ^  ±  i  /  ^  "f"  X  ^^^^^^  which  do  not  dif- 
fer from  the  preceding,  except  in  the  sign.  Thus  the  same 
equation  connects  together  two  questions,  which  differ  from  each 
other  only  in  the  sense  of  certain  conditions. 

2.  Again,  let  Q  be  negative.     The  equation  will  then  be  of 
the  form  2r^  db  Pa;  =  —  Q,  and  we  have 


Here,  in  or(Jer  that  the  root  of  the  quantity  placed  under  the 
radical  sign  may  be  taken,  or  in  other  words,  that  the  value  of  x 

F 

may  be  real,  it  is  evident  that  Q  must  not  exceed  -^ . 

/p  p 

Since  moreover  \y/   -r Q  is  numerically  less  than  —  it 

follows,  that  the  values  of  x  will  both  be  negative,  if  P  is  posi- 
tive in  the  equation,  that  is,  if  the  equation  is  of  the  form 
3?-i^Vx  =  —  Q,  and  that  they  wdll  both  be  positive,  if  P  is 
negative  in  the  equation,  that  is,  if  the  equation  is  of  the  form 
ar^  — Pa;  =  — Q. 

Indeed,  it  may  be  shown  a  priori,  that  always  tvJieii  Q  is 
negative,  in  the  second  memher  and  P  negative  in  the  first,  the 
problem  will  admit  of  tivo  direct  solutions,  provided  that  Q  does 

not  exceed  -v-. 
4 

The  equation  x^  —  Pa:  =  — Q,  may,  by  changing  the  signs 
of  all  the  terms,  be  put  under  the  form 

Pa:  — r'  =  Q,  or  a:  (P  —  a:)  =  Q. 

But  the  equation  a:  (P  —  a:)  =  Q  is  evidently  the  algebraic 
translation  of  the  following  enunciation,  viz.  To  divide  a  number 


154  ELEMENTS    OF    ALGEBRA. 

P  into  iiDo  parts,  the  product  of  which  shall  he  equal  to  a  given 
number  Q.  For  if  we  put  x  for  one  of  the  parts,  the  other 
part  will  be  P  —  x,  and  the  product  of  the  two  parts,  will  be 
x{V  —  x). 

This  being  premised,  the  enunciation  of  the  problem  admits, 
it  is  evident,  of  two  direct  solutions;  for  the  equation  of  the 
problem  will  be  the  same,  whether  x  be  put  for  one  or  the  other 
of  the  parts ;  there  is  no  reason  then,  why  the  equation,  when 
resolved,  should  give  one  of  the  parts  rather  than  the  other ;  it 
should  therefore  give  both  at  the  same  time. 

Moreover,  in  order  that  the  problem  may  be  possible,  it  is 

p2 

necessary,  that  Q  should  not  exceed  ~ ;  for  the  greatest  possible 
product  of  the  parts,  into  which  the  number  P  may  be  divided 

P2     ,      . 

being  equal  only  to  —  it  is  absurd  to  require  that  their  product, 

P 

which  we  have  represented  by  Q,  should  be  greater  than  — .     We 

conclude  therefore  that,  in  all  cases  when  the  hnoion  quantity 
is  negative  in  the  second  member,  but  numerically  greater  than 
the  square  of  half  the  coefficient  of  the  second  term,  the  question 
proposed  is  impossible. 

117.  The  following  examples  will  serve  as  an  exercise  upon 
the  different  cases,  which  we  have  here  been  considering.  What 
change  must  be  made  in  the  enunciations  of  the  first  four  ques- 
tions respectively,  in  order  that  the  negative  solutions  may  become 
positive  ?  How  must  the  fifth  question  be  modified,  so  that  the 
answers  shall  become  positive  ?  In  what  does  the  absurdity  in 
the  seventh  question  consist  ? 

1.  A  company  at  a  tavern  had  £8,  15^.  to  pay,  but  two  of 
them  having  left  before  the  bill  was  settled,  those  who  remained 
had  each  in  consequence  IO5.  more  to  pay.  How  many  were  in 
the  company  at  first  ? 

2.  A  man  travelled  105  miles,  and  then  found  that  if  he  had 
not  travelled  so  fast  by  2  miles  an  hour,  he  should  have  been 


GENERAL   EQUATION   OF   THE    SECOND    DEGREE.  155 

6  hours  longer  in  performing  the  same  journey.     How  many- 
miles  did  he  go  per  hour  ? 

3.  A  regiment  of  foot  was  ordered  to  send  216  men  on  gar- 
rison duty,  each  company  being  to  furnish  an  equal  number; 
but  before  the  detachment  marched,  3  of  the  companies  were  sent 
on  another  service,  when  it  was  found  that  each  company  that 
remained  was  obliged  to  furnish  12  additional  men,  in  order  to 
make  up  the  complement  216.  How  many  companies  were 
there  in  the  regiment,  and  what  number  of  men  was  each  ordered 
to  send  at  first  ? 

4.  A  and  B  set  out  from  two  towns,  which  were  distant  247 
miles,  and  travelled  the  direct  road  till  they  met.  A  went  9 
miles  a  day ;  and  the  number  of  days  at  the  end  of  which  they 
met,  was  greater  by  3,  than  the  number  of  miles,  which  B  went 
in  a  day.     Where  between  A  and  B  did  they  meet  ? 

On  substituting  —  x  for  x  in  the  equations,  which  pertain 
respectively  to  the  preceding  questions,  it  will  be  easy  to  trans- 
late these  equations  into  enunciations  analogous  to  those  of  the 
questions  proposed ;  there  are  questions  however,  in  which  it  will 
be  very  difficult  to  do  this,  and  the  negative  solutions  in  such 
cases  are  to  be  regarded  merely  as  connected  with  the  first  in  the 
same  equation  of  the  second  degree. 

5.  A  gentleman  counting  the  guineas,  which  he  had  in  his 
purse,  finds  that  if  24  be  added  to  their  square,  and  8  times  their 
number  be  subtracted  from  17,  the  sum  and  remainder  will  be 
equal.     How  many  guineas  had  he  in  his  purse  ? 

6.  A  set  out  from  C  towards  D,  and  travelled  7  miles  a  day. 
After  he  had  gone  32  miles,  B  set  out  from  D  towards  C,  and 
went  every  day  one-nineteenth  of  the  whole  journey ;  and  after 
he  had  travelled  as  many  days  as  he  went  miles  in  one  day,  he 
met  A.     Required  the  distance  of  the  places  C  and  D  ? 

7.  The  difference  of  two  numbers  is  7,  and  the  square  of  the 
greater  is  equal  to  25  times  the  less.     What  are  the  numbers  f 


156  ELEMENTS    OF   ALGEBRA. 


EXAMINATION   OF    PARTICULAR   CASES. 

1.  In  the  general  equation  let  Q  be  negative,  that  is,  let  the 
equation  be  of  the  form  x^-\-Fx  =  —  Q,  P  being  of  any  sign 

whatever;  if  we  suppose  Q  =  — ,  the  radical 


\/ 


pa 


will  be  reduced  to  0,  and  the  values  of  x  will  be  equal  each  to 

P  F 

—  — .     Thus  if  Q  be  negative  in  the  equation  and  equal  to  -^, 

the  values  of  x  will  be  equal,  and  will  both  be  positive  if  P  is 
negative,  or  both  negative  if  P  is  positive. 

2.  In  the  general  formula 

let  Q  =  0,  the  values  of  x  will  then  be  a:  =  0,  a;  =  —  P. 

3.  In  the  same  formula  let  P  =  0,  we  have  then 

that  is  to  say,  the  values  of  x  will  in  this  case  be  equal,  but  of 
contrary  signs,  real  if  Q  is  positive,  and  imaginary  if  Q  is 
negative. 

4.  Let  P  =  0,  Q  =  0,  the  values  of  x  will  then  be  each  equal 
to  0. 

5.  We  have  next  to  examine  a  remarkable  case  which  fre- 
quently occurs  in  the  solution  of  problems  of  the  second  degree. 
For  this  purpose,  let  us  take  the  equation 

Aar^+Ba:=C. 

This  equation  being  resolved,  gives 

—  B±VB'^  +  4AC 

^= ^ ' 

Let  it  now  be  supposed,  that  in  consequence  of  a  particular 

hypothesis  made  upon  the  given  things  in  the  question,  we  have 

A=  0,  the  values  of  x  then  become 

_0       _      2B 


DISCUSSION   OF   PROBLEMS   OF    THE    SECOND   DEGREE.  157 

The  second  value  of  x  here  presents  itself  under  the  form  of 
infinity,  and  may  be  regarded  as  a  true  answer,  when  the  ques- 
tion is  susceptible  of  infinite  solutions.  In  order  to  interpret  the 
first,  if  we  return  to  the  equation,  we  see  that  the  hypothesis 

Q 

A  =  0  reduces  it  to  Ba:  =  C,  from  which  we  deduce  x  =  ^,  an 

expression  Jinite  and  determinate,  and  which  must  be  regarded 

as  the  true  value  of  ^  in  the  present  case. 

6.  Let  it  be  supposed  finally,  that  we  have  at  the  same  time 
A  =  0,  B  =  0,  C  =  0.  The  equation  will  then  be  altogether 
indeterminate.  This  is  the  only  case  of  indetermination,  which 
the  equation  of  the  second  degree  presents. 


DISCUSSION    OF    PROBLEMS. 

118.  The  following  problems  offer  all  the  circumstances,  which 
usually  occur  in  problems  of  the  second  degree. 

1.  To  find  on  the  line  A  B,  which  joins  two  luminous  bodies 
A  and  B,  the  point  where  these  bodies  shine  with  equal  light. 


C"  A  C  B  C 

The  solution  of  this  problem  depends  upon  the  following  prin- 
ciple in  physics,  viz.  The  intensity  of  light  from  the  same 
luminous  body  will  be,  at  different  distances,  in  the  inverse  ratio 
of  the  square  of  the  distance. 

This  being  premised,  let  <z  =  A  B,  the  distance  between  the 
two  bodies ;  let  b=  the  intensity  of  A  at  the  unit  of  distance, 
c  =  the  intensity  of  B  at  the  same  distance ;  let  C  be  the  point 
required,  and  let  A  C  =  x. 

The  intensity  of  A  at  the  distance  1  being  b,  its  intensity 

&t  the  distance  2,  3,  4,  ...  .  will  ^  t>  q>  t^  •   • .  •   • .  and  by 
consequence,  at  the  distance  x,  it  will  be  -5.    For  the  same 


158 


ELEMENTS   OF   ALGEBRA. 


reason,   the   intensity    of    B    at   the    distance   a  —  x  will  be 
;  whence,  by  the  question,  we  have  i 


{a  —  x)' 


From  which,  we  obtain 


{a^xf 


a'b 


cY 


b  —  c 


or  reducing 


a{b±^bc) 


But^±/v/^c  may,  it  will  be  observed,  be  put  under  the 
form  /^  b  (^  b  zLa/  c),  and  b  —  c  may  be  put  under  the  form 
{A/bf-i^cf,  or  {^b  +  ^c)  i^b-^c). 

Taking  advantage  of  this  remark,  the  value  of  x  may  be 
expressed  more  simply,  thus 

X  =  — J-. J-,  whence  a  —  x=  —— — ^'^- 


or         x  = 


a  \f  b 


V^  +  Vc 
a  s/  b 


■  and 


a  /s/  c 


DISCUSSION. 

1.  Let  b  be  greater  than  c. 

The    first    value    of   x   is    positive    and   less   than  <z,   since 

.  ,     - — J-  is  a  fraction.     The  point  sought,  therefore,  accord- 

ing  to  this  value  of  x,  is  situated  between  A  and  B.  It  is 
moreover  nearer  B  than  A ;  for,  in  consequence  of  3  ^  c, 
we    have    ^  b  -{-  /s/  b,    or    '^  h/  b^  s/  h  -\-  s/  c,    whence 

aJ  b  1  a/s/  b       ^  a 

17b +  ^0 > 2'  ''"'*  ^y  consequence  ^^^^^ >^. 

This,  indeed,  should  be  the  case,  since  we  have  supposed  the 
intensity  of  A  greater  than  that  of  B. 

The  corresponding  value  of  a  —  x  is  also  positive  and  less 
as  it  will  be  easy  to  see,  than  -. 


DISCUSSION    OF    PROBLEMS    OF    THE    SECOND    DEGREE.  159 

The  second  value  of  x  is  positive,  but  greater  that  c,  since 

we  have      .  ,    -z-  ">  1.     This  value  of  x  gives,  therefore,  a 

V  0  —  f^  c 

second   point  C  situated  upon  AB  produced  and  at  the  right 

of  A  and  B.     Indeed,  since  the  light  from  A  and  B  expands 

itself  in  all  directions,  there  should  be,  it  is  easy  to  see,  on  AB 

produced  a  second  point  where  A  and  B  shine  with  equal  light. 

This  point  moreover  should  be  nearer  the  body,  the  light  of 

which  is  least  intense. 

The  second  value  oi  a  —  a:  is  negative :  this  should  be  the  case, 

since  we  have  x^a. 

2.  Let  b  be  less  than  c. 

a 
The  first  value  of  x  is  positive,  but  less  than  -.     The  corre- 
sponding value  of  a  —  a;  is  also  positive  and  greater  than  -. 

Thus  on  the  present  hypothesis  the  point  C,  situated  between 
A  and  B,  should  be  nearer  A  than  to  B. 

The  second  value  of  x  is  essentially  negative.     In  order  to 

interpret  it,  we  return  to  the  equation,  which  becomes  by  substi- 

b  c 

tuting — a:  for  a:, -^  = -— -j— — -^.     But  a  —  x  expressing  in   the 

X  ( Q*  ~T~  Xj 

first  instance  the  distance  of  the  point  sought  from  B,a-\-  x  must 
in  the  present  case  express  the  same  distance.  Thus  the  point 
sought  should  be  at  the  left  of  A,  in  C"  for  example.  Indeed, 
since  by  hypothesis  the  intensity  of  B  is  greater  than  that  of  A, 
the  second  point  sought  should  be  nearer  A  than  to  B. 

3.  Let  b  =  c. 

The  first  value  of  x,  and  also  that  of  a  —  z  is  reduced  in  this 

case  to  -.  Thus  we  have  the  middle  of  AB  for  the  point  sought. 
This  result  conforms  to  the  hypothesis. 

The  remaining  values  are  reduced  to  -^ —  or  become  infinite, 
that  is,  the  second  point  where  the  bodies  shine  with  equal  .light, 


160  ELEMENTS    OF   ALGEBRA. 

is  situated  at  a  distance  from  A  and  B  greater  than  any  assign- 
able quantity.  This  result  corresponds  perfectly  with  the  present 
hypothesis;  for,  if  we  suppose  the  difference  b  —  c,  instead 
of  being  absolutely  nothing,  to  be  very  small,  the  second  point 
will  exist,  but  at  a  very  great  distance  from  A  and  B.  If  then 
b  =  c  or 

the  point  required  must  cease  to  exist,  or  be  placed  at  an  infinite 
distance. 

4.  Let  b  =  c  and  a  =  0. 

The  first  system  of  values  of  x  and  a  —  x  reduce  themselves 

in  this  cat*  to  0,  and  the  second  system  to  ^.  This  last  charac- 
ter is  here  the  symbol  of  indetermination ;  for,  on  returning  to 
the  equation  of  the  problem, 

[b  —  c)  a^ — 2abx  =  —  fl^3,  • 

this  equation  becomes  on  the  present  hypothesis 

0.2^—0.  ar=-0, 
an  equation  which  may  be  satisfild  by  any  number  whatever 
taken  for  x.     Indeed,  since  the  two  bodies  have  the  same  intensity 
and  are  placed  at  the  same  point,  they  should  shine  with  equal 
light  upon  any  point  whatever  in  the  line  A  B. 

5.  Finally  let  a  =  0,  b  being  different  from  c. 

Both  systems  in  this  case  will  be  reduced  to  0,  which  in- 
dicates, that  there  is  but  one  point,  where  the  bodies  shine 
with  equal  light,  viz.  the  point,  in  which  the  two  bodies  are 
situated. 

2.  To  find  two  numbers  such,  that  the  diflierence  of  their 
products  by  the  numbers  a  and  b  respectively  may  be  equal  to  a 
given  number  s,  and  the  difference  of  their  squares  equal  to 
another  given  number  q. 

Denoting  by  x  and  y  the  numbers  sought,  we  have  by  the 
question  ax  —  by  =  s 

x'  —  f^g. 


DISCUSSION    OF    PROBLEMS    OF    THE    SECOND    DEGREE.  161 

Resolving  these  equations,  we  have  for  the  first  system  of 
values  for  x  and  y 


as-\-l 

^V^- 

-Q{^- 

-b-) 

a^- 

-b^ 

_bs  +  a^  ^- 

-q{d^- 

-f>^) 

d'- 

-h' 

and  for  the  second  system,  we  have 


as- 

-bVs"- 

-q{d'- 

-b^) 

d"- 

-b^ 

bs- 

-a\f  s"- 

-q{d^- 

-*'1 

^"^  d'—b^ 


DISCUSSION. 


1.  Let  a  be  greater  than  b,  and  by  consequence  a* — b"^  posi- 
tive 

In  order  that  the  values  of  x  and  y  may  be  real,  it  is  necessary 
that  we  have 

q  {c^ ' —  b^)  <^  s^,  and  therefore,  q  <^  -^ jz. 

This  condition  being  fulfilled,  the  values  of  x  arid  y  in  the  first 
system  will  be  necessarily  positive,  and  will,  by  consequence, 
form  a  direct  solution  of  the  problem  in  the  sense,  in  which  it  is 
enunciated. 

In  the  second  system  the  value  of  x  will  be  essentially  posi- 
tive; fora]>^  gixes  as'^bs,  and  for  a  still  stronger  reason, 
as^bs/s'—q{d^  —  b''). 

With  respect  to  the  value  of  y^  it  may  be  either  positive  or 
negative.     In  order  that  it  may  be  positive,  we  must  have 

bs^aVs'  —  qid'—b^) 
or,  squaring  both  sides, 

11 


162  ELEMENTS    OF    ALGEBRA. 

or,  adding  c^  q  [a^  —  //)  to  both  sides  of  this  last,  and  subtracting 
b^s'  from  both  sides 

or,  by  division  S'^l- 

Thus,  in  order  that  the  second  system  maybe  a  real  and  direct 
solution,  we  must  have 

If,  then,  we  take  for  a,  b,  and  5  any  absolute  numbers  what- 
ever, provided  that  we  have  a'^b  and  that  we  take  for  q  a 

number  comprised  between  the  two  limits  -^  and  -^ r^,  we  shall 

(T  (T  —  b^ 

be  certain  of  obtaining  two  direct  solutions. 

Thus,  let  a  =  6,  ^  =  4,  5  =  15 ;  we  have 

i       225      „,        ,       /  225       „. 

if  then  we  take  q  ==  10,  for  example,  we  shall  have 
6X  15±4V225— 20X  10       11       7 


^-                       20 

-2   ^^2 

4Xl5±6/v/225- 

-20X10       9        3 

^—                       20 

2'  °'  2' 

If  on  the  present  hypothesis,  we  have  ?  <C^  — >  and  for  a  still 

stronger  reason,  q  <C  —^ JIS'  ^^®  value  of  y  in  the  second  sys- 
tem will  be  negative.  This  system,  therefore,  will  not  be  a 
solution  of  the  proposed  problem  in  the  sense,  in  which  it 
is  enunciated,  but  of  an  analogous  problem,  the  equations  of 
which  are  ax-{-by  =  s 
^  3?  —  f  =  q 

and  which  will  differ  from  the  proposed  in  this  respect  only,  that 
s  will  express  an  arithmetical  sum  instead  of  difference. 
2.  Let  a  be  less  than  b  and  therefore  c^  —  b"^  negative. 


DISCUSSION    OF   PROBLEMS    OF    THE    SECOND   DEGREE.  163 

In  this  case  the  expressions  for  x  and  y  in  the  first  system 
may  be  put  under  the  form 

_  —  as  —  hs/^-\-q{}i'^d') 
''""  h'  —  d' 

__  —  bs  —  a^/s-  -j-q{l?^-—  a') 
and  in  the  second 

'^~"  b-'  —  a' 
—  bs^a  s/s"  -\-q{b'  —  a') 

y—  ^537^  • 

The  values  of  x  ,and  y  in  both  systems,  it  is  evident,  will  be 
real,  since  the  quantity  placed  under  the  radical  is  essentially 
positive. 

In  the  first  system  the  values  of  x  and  y  are  essentially  nega- 
the;  in  the  second  the  value  of^a:,  it  is  easy  to  see,  is  neces- 
sarily positive,  but  the  value  of  y  may  be  either  positive  or  nega- 

tive ;  in  order  that  it  may  be  positive,  we  must  have  q^—^. 

3.  Let  a-=hj  and  therefore  d^  —  b'^=.  0. 
On  this  hypothesis,  we  have  for  the  first  system  of  values 
for  x  and  y 

2as  2as 

^=-0-.  y=^; 

and  for  the  second 

0  0 

Returning  to  the  equations  of  the  proposed  in  order  to  inter- 
pret these  last,  we  obtain  for  x  and  y  on  the  present  hypothesis 


?q-\-s^       ^^Q — ^ 

2as    '  ^~"     2as    ' 


PROBLEMS    FOR    SOLUTION    AND    DISCUSSION. 

1.  There  are  two  numbers,  whose  sum  'is  a  and  the  sum  of 
whose  second  powers  is  b.     Required  the  numbers. 


164  ELEMENTS    OF   ALGEBRA. 

futtijig  z  and  y  for  the  numbers,  we  have 


a^s/2h- 

-d' 

~~             2 

azh^2b- 

-a^ 

What  conditions  are  necessary  in  order  that  the  values  of  x 
and  y  may  be  real  ?  When  will  the  values  of  x  both  be  posi- 
tive? Can  both  be  negative?  When  will  one  of  them  be 
positive  and  the  other  negative,  and  to  what  question  does  the 
negative  value  belong  ? 

2.  To  find  two  numbers  such,  that  the  sum  of  their  products 
by  the  numbers  a  and  b  respectively  may  be  equal  to  25,  and 
their  product  equal  to  p. 

Putting  X  and  y  for  the  numbers,  we  have 

szh^s^  —  abp 

x=. — 

a 

szh^s^  —  abp 

y  = -^ . 

What  conditions  are  necessary  in  order  that  the  values  of  x 
and  y  may  be  real  ?  What  is  the  greatest  value  of  which  p  ad- 
mits ?     Can  either  of  the  values  of  a:  or  y  be  negative  ? 

8.  To  find  two  numbers  such,  that  the  sum  of  their  products 
by  the  numbers  a  and  b  respectively  may  be  equal  to  a  given 
number  s,  and  the  sum  of  their  squares  equal  to  another  given 
number  q. 

Putting  X  and  y  for  the  numbers  respectively,  we  have 
__as±b  ^{d'  +  b') q  —  s" 

_bszt^^{d'  +  l>')q  —  s' 
y—  a^J^b^ 

What  conditions  are  necessary  in  order  that  the  values  of  x 
and  y  may  be  real  ?  Within  what  limits  must  q  be  comprised 
\\i,  order  that  both  values  of  x  may  be  positive  ?  Within  what 
limits  must  q  be  comprised  in  order  that  both  values  of  y  may 


JiAXlMA  AND  IttlNIMA.  165 

he  positive?  In  the  second  system  of  values  for  x  and  y, 
when  will  the  value  of  x  be  positive,  and  that  of  y  negative, 
and  what  is  the  analogous  problem,  to  which  this  system  be- 
longs ? 

4.  To  find  a  number  such,  that  its  square  may  be  to  the 
product  of  the  differences  between  this  number  and  two  other 
numbers  a  and  b  in  the  ratio  of  q  to  p. 

Putting  X  for  the  number  sought,  we  have 

q{a  +  b)  ±  ^q'  {a  —  bf  +  4:pqab 
'^-  2{q-^p) 

Let  this  formula  be  examined  on  the  different  hypotheses 
9<P^  S=P,  Q>P' 


SECTION  XIV.— Maxima  and  Minima. 

119.  In  several  of  the  preceding  questions,  the  given  things, 
we  have  seen,  are  so  connected  among  themselves,  that  one  is 
determined  by  the  others  to  be  comprised  within  certain  limits, 
or  to  have  a  greatest  or  least  possible  value. 

A  quantity,  the  value  of  which  may  be  made  to  vary,  is  called 
a  variable  quantity;  the  greatest  value  of  which  is  called  a 
viaximum  and  the  least  a  minimum. 

Questions  frequently  occur,  in  v/hich  it  is  required  to  deter- 
mine under  what  circumstances  the  result  of  certain  arithmet- 
ical operations  performed  upon  numbers  will  be  the  greatest  or 
least  possible.  We  shall  resolve  a  few  questions  of  this  kind 
the  solutions  of  which  depend  upon  equations  of  the  second 
degree. 

1.  To  divide  .a  number,  2a,  into  two  parts  such,  that  the 
product  of  these  parts  may  be  a  maximum. 

Let  X  be  one  of  the  parts,  then  2a  —  x  will  be  the  other,  and 
their  product  will  be  x{2a  —  x).     By  assigning  different  values 


166  ELEMENTS    OF    ALGEBRA. 

to  x^  the  product  xi^a  —  x)  will  vary  in  magnitude,  and  the 
question  is  to  assign  to  :?:  a  value  such,  that  this  product  may  be 
the  greatest  possible.  Let  m  be  the  maximum  sought,  we  have 
by  the  question 

Eegarding  for  the  moment  m  as  known,  and  deducing  from 
this  equation  the  value  of  x^  we  have 

a:  =  a  ±  w  of-  —  m.         * 

From  this  result  it  appears,  that  in  order  that  x  may  be  real, 
m  must  not  exceed  c? ;  the  greatest  value  of  m  will  therefore  be 
c^^  in  which  case  we  have  x  =  a.  Thus  to  obtain  the  greatest 
possible  product,  the  proposed  must  be  divided  into  two  equal 
parts,  and  the  maximum  obtained  ivill  be  equal  to  the  square  of 
one  of  these  parts. 

In  the  equation  x{2a  —  x)=^m,  the  expressions  x{2a  —  x) 
is  called  a  function  of  x.  This  function  is  itself  a  variable, 
the  value  of  which  depends  upon  that  given  to  the  first  variable 
ox  X. 

2.  To  divide  a  number,  2a,  into  two  parts  such,  that  the  sum 
of  the  square  roots  of  these  parts  may  be  a  maximum. 

Let  7?  be  one  of  the  parts,  then  2  a  —  x^  will  be  the  other,  and 
the  sum  of  the  square  roots  will  be  x-\r  ^ 2a  —  x^.  Let  m  be 
the  maximum  sought,  we  have  by  the  question 

x-\-^2a  —  x^=-m;  ^'—<^Jn  ^  ^' 


2a 


from  which  we  obtain     .'^-  '^  ''  f      s   •''^^'^   — •     -  ■  ^^'^^ 

^   ,    4     /  "^ 
''=2^V    ^ 

or  simplifying  a;  =  ^  ±  _  ^-^^ZT^, 

In  order  that  the  values  of  a:  may  be  real,  the  value  of  m^  must 
not  exceed  4a;  2 /^  a  is  therefore  the  greatest  value,  which  wi 
can  receive. 

Let  us  put  m==-2 s/ a,  we  have  x==- tsj a  and  3?  =  a,  whence 


i 


-f-   4./ 


MAXIMA    AND   MINIMA.  167 

2a  —  x^  =  a.  Thus,  the  proposed  must  be  divided  into  two  equal 
parts  in  order  that  the  sum  of  the  square  roots  of  the  parts  may 
be  a  maximuyn.  This  maximum  moreover  will  be  equal  to  twice 
the  square  root  of  one  of  the  parts. 

From  what  has  been  done,  the  following  rule  for  the  solution 
of  questions  of  the  kind  which  we  are  here  considering  will 
readily  be  inferred^yi-z.  HarAng  formed  the  algebraic  expression 
of  the  quantity  f^jeptible  of  becoming  a  maximum  or  minimum, 
make  this  expression  equal  to  any  quantity  m.  If  the  equation 
thus  made  is  of  the  second  degree  in  x,  x  designating  the  variable 
quantity^  lohich  enters  into  the  algebraic  expression^  resolve  this 
equation  in  relation  to  x;  make  next  the  quantity  under  the 
radicaLequal  to  zero,  and  deduce  from  this  last  equation  the  value 
ofm;  this  ivill  be  the  maximum  or  minimum  sought.  Substi- 
tuting finally  the  value  of  m  in  the  expressioii  for  x,  lue  obtain 
the  value  of  x  proper  to  satisfy  the  enunciation  proposed. 

If  the  quantity  placed  under  the  radical  remains  essentially 
positive,  whatever  the  value  of  m,  Vv^e  infer  that  the  expression 
proposed  may  be  of  any  assignable  magnitude  whatever,  or  in 
other  words,  that  it  will  have  infinity  for  a  maximum  and  zero 
for  a  minimum. 

42-4- 4 :r 3 

Thus  let  there  be  proposed  the  expression       c/o.     i~T\ — ' 

to  determine  v/hether  this  expression  is  susceptible  of  a  maxi- 
mum or  minimum. 

4^2   I   4^. 3 

Putting      i^i^'  ,   I    -  >^     =  ^  and  deducing  the  value  of  a:,  we 

v)  I  ^  *//    '   I  ■    Jl  I 

3^  —  1   ,   1    , 

have  X  = g ^-2^9m'-\-^.     Here,  whatever  value  we 

give  to  m,  the  quantity  placed  under  the  radical  will  be  positive ; 
the  proposed  therefore  may  be  of  any  magnitude  whatever. 

EXAMPLF'-S    FOR    PRACTICE. 

1.  To  divide  a  given  number  a  into  two  factors,  the  sum  of 
which  shall  be  a  minimum. 

Ans.  The  two  factors  should  be  equal. 


168  ELEMENTS    OF    ALGEBKA. 

2.  Let  d  be  the  difference  between  two  numbers;  required 
that  the  square  of  the  greater  divided  by  the  less  may  be  a 
minimum.       Ans.  The  minimum  required  is  4^  and  the  value 

of  the  greater  part  corresponding  is  2d. 

3.  Let  a  and  b  be  two  numbers  of  w^hich  a  is  the  greater,  to 
find  a  number  such,  that  if  a  be  added  "to  this  number,  and  b  be 
subtracted  from  it,  the  product  of  the  suny-^nd  difference  thus 
obtained  being  divided  by  the  square  of  the  number,  the  quotient 
will  be  a  maximum. 


nnn  thc»   m«svinrmm    J ' 


Ans.  The  number  =:  '^'^'^,,  and  the  maximum  .     , 

a  —  b  ^ab 


4.  To  divide  a  number  2  a  into  two  parts  such,  that  the  sum 
of  the  quotients  obtained  by  dividing  the  parts  mutually,  one  by 
the  other,  may  be  a  minimum.    ' 

Ans.  The  number  should  be  divided  into  two 
equal  parts,  and  the  minimum  is  2. 

5.  To  find  a  number  such,  that  if  a  and  b  be  added  to  this 
number  respectively,  the  product  of  the  two  sums  thus  obtained, 
divided  by  the  number,  may  be  a  minimum. 

Ans.    The  number  =  /^y  ab,  and  the  minimum 


SECTION  XV. — Powers  and  Roots  of  Monomials. 

120.  AVhen  a  quantity  is  multiplied  into  itself,  the  product, 
we  have  seen,  is  called  a  poiver,  the  degree  of  which  is  marked 
by  the  exponent  of  tne  product,  thus  aaaaa  or  a^  is  called  the 
f^fih  power  of  a;  in  like  manner  o/'  is  called  the  mih.  power 
of  a. 

The  original  quantity,  from  which  a  power  is  derived,  is 
called  the  root  of  this  power.  The  degree  of  the  root  is  de- 
termined by  the  number  of  times  the  root  is  found  as  a  factor 
in  the  poorer ;  thus  a  is  the  fifth  root  of  a^ ;  in  like  manner  a 


POWERS   AND   ROOTS    OF   MONOMIALS.  169 

is  the  ?7zth  root  of  a"*.     The  number  which  marks  the  degree  of 
the  root  is  called  the  index  of  the  root. 

121.  Let  it  be  proposed  to  find  the  fifth  power  of  2a^3'; 
this  power  is  indicated  thus,  {2a?b'^f,  and  we  have,  it  is  evi- 
dent, 

{2a'by  =  2a'b^  X  ^a'b^  X  ^a^b^  X  ^ciH''  X  2(^b\ 

Here,  it  is  evident,  1°.  that  the  coefficient  2  must  be  multiplied 
into  itself  four  times  or  raised  to  the  fifth  power ;  2°.  that  each 
one  of  the  exponents  of  the  letters  must  be  added,  until  it  is 
taken  as  many  times  as  there  are  units  in  the  exponent  of  the 
power,  or  in  other  words,  multiplied  by  5 ;  we  have  therefore 
{2a^b''f=:32a''b'', 

In  like  manner  {8aH'cY  =  5l2a^b'c\ 

To  raise  a  monomial  therefore  to  any  given  power,  we  raise 
the  coefficient  to  this  potver,  and  multiply  each  one  of  the  expo^ 
nenis  of  the  letters  by  the  exponent  of  the  povjer. 

With  respect  to  the  sign,  with  which  the  powers  of  a  mono- 
mial should  be  affected,  it  is  evident,  that  whatever  be  the  sign 
of  the  quantity  itself,  its  second  power  will  be  positive.  More- 
over if  the  exponent  of  the  power  of  a  monomial  .,]be  an  even 
number,  it  is  easy  to  see,  that  this  power  may  be  considered 
as  a  power  of  the  square  of  the  proposed  quantity.  Thus  a^,  it 
is  evident,  may  be  considered  as  the  fourth  power  of  a^ ;  in  like 
manner  a^"",  any  even  power  of  a,  may  be  considered  the  mih 
power  of  a^.  It  follows,  therefore,  that  lohatever  be  the  sign  of  a 
monomial,  any  poiver  of  it,  the  exponent  of  ivhich  is  an  even 
number,  is  positive. 

Again,  since  the  power  of  a  simple  quantity,  the  exponent 
of  which  is  an  odd  number,  is  equal  to  a  power  of  this  quan- 
tity of  an  even  degree  multiplied  by  the  first  power,  it  follows, 
that  every  power  of  a  monomial,  the  exponent  of  which  is  an  odd 
number,  loill  have  the  same  sign  as  the  quantity  from  ivhich  it  is 
formed. 

122.  Let  it  now  be  proposed  to  find  the  third  root  of  64a'i'c'. 

o 


170  ELEMENTS    OF    ALGEBRA. 

The  root  required  is  indicated  thus,  wQ'la'^b^c^;  the  sign  a/ 
being  employed  to  denote  in  general  that  a  root  is  to  be  taken, 
and  the  index  3.  placed  above  the  radical  sign  to  denote  the 
particular  root  required. 

Since  the  root  of  a  quantity  must  evidently  be  sought  by  a 
process  the  reverse  of  that,  by  which  it  is  raised  to  a  power,  in 
order  to  extract  the  root  of  a  monomial,  1°.  we  extract  the  root 
of  the  coefficient,  2°.  we  divide  the  exponent  of  each  of  the  letters 
hy  the  index  of  the  root. 

According  to  this  rule,  the  third  root  of  the  proposed  will 
be  ^c^h^c.  In  like  manner  the  seventh  root  of  a^^b^^c^  is 
aH'c\ 

With  respect  to  the  signs,  with  which  the  roots  of  monomials 
should  be  affected,  it  is  an  evident  consequence  of  the  principles 
already  established  that, 

1°.  Every  root  of  an  even  degree  of  a  positive  monomial  may 
have  indifferently  either  the  sign  -j-  or  — .  Thus,  the  sixth  root 
of  64«^Ms±2a^ 

2°.  Every  root,  the  degree  of  which  is  expressed  by  dn  odd 
number,  will  have  the  same  sig7i  as  the  quantity  proposed.  Thus 
the  fifth  roet  of  —32a'' b'  is  —2a'b. 

3°.  Every  root  of  an  even  degree  of  a  negative  monomial  is  an 
impossible  or  imaginary  root.  For  there  is  no  quantity,  which 
raised  to  a  power  of  an  even  degree  can  give  a  negative  result. 

Thus  s/ —  a,  \/ —  b  denote  impossible  or  imaginary  quanti- 
ties, in  the  same  manner  as  'v  —  a,  V  —  b. 

123.  From  what  has  been  said,  it  is  evident  in  order  that  a 
root  may  be  extracted,  1°.  that  the  coefficient  of  the  proposed 
must  be  a  perfect  power  of  the  degree  marked  by  the  index  of 
the  root  to  be  extracted ;  2°.  that  the  exponents  of  each  of  the 
letters  must  be  divisible  by  the  index  of  the  root. 

When  this  is  not  the  case  the  root  can  only  be  indicated.  It 
should  be  observed,  however,  that  radical  expressions,  of  any 
degree  whatever  admit  of  the  same  simplifications  as  those  of  the 


THEORY    OF    COMBINATIONS.  .  171 

second  degree.  These  simplifications  are  founded  upon  the  prin- 
ciple, that  any  root  whatever  of  a  product  is  equal  to  the  product 
of  the  same  root  of  the  several  factors. 

Thus  let  it  be  proposed  to  find  the  third  root  of  SAa^Pc^. 
The  third  root,  it  is  evident,  cannot  be  taken ;  for  54  is  not  a 
perfect  third  power,  and  the  exponents  of  the  letters  a  and  c 
are  not  divisible  by  3.  We  therefore  indicate  the  root,  thus, 
w54.a*l)^c^;  but  this  expression  may  be  put  under  the  form 
^27^^^*^  X  2<zr;  whence  taking  the  third  root  of  the  factor 

27an\  we  have  W5UF¥?=SabW2^. 
Let  it  be  proposed  next  to  find  the  third  root  of 
l25a'b-{-S75a^c. 
This  expression,  which  it  is  easy  to  see  is  not  a  perfect  third 
power,  may  be  put  under  the  form  125  a^  [a^  b -\- 2  c) ;  whence 
extracting  the  third  root  of  the  first  factor,  we  have  for  the  root 

sought  5  a  w  ctb  -j-  3  c. 

EXAMPLES. 

1.  To  reduce  ^/S^a^b''  c  to  its  most  simple  form. 

2.  To  reduce  ^/l^Sx^ifz^  to  its  most  simple  form. 

3     - • 

3.  To  reduce  wax^  -\-bx^  io  its  most  simple  form. 

4.  To  reduce  ^\/21  d^  -\- SI  a^b  to  its  most  simple  form. 

5.  To  reduce  V96a^^'c^^  to  its  most  simple  form. 

^    rp        T        'i     /md'  —  \Qa'b'       .  .      ,    ^ 

b.   lo  reduce   1  /     r-—- ^^  ^  ^   , ,  to  its  most  simple  form. 

Y       SW  —  324a*  ^  ^ 


SECTION  XVI. — Powers  of  Compound  Quantities,  Theory 
OF  Combinations,  Binomial  Theorebl 

124.  Powers  of  compound  quantities  are  found  like  those  of 
monomials  by  the  continued  multiplication  of  the  quantity  into 
itself.     They  are  indicated  by  inclosing  the  quantity  in  a  paren- 


172  ELEMENTS   OF   ALGEBRA. 

thesis,  to  which  is  annexed  the  exponent  of  the  power.  The 
third  power  of  a^-\-5ab  —  bc^,  for  example,  is  indicated,  thus, 
{a^-\-5ab  —  bc^f.  This  same  power  may  also  he  indicated 
thus,  d^-\-5ab  —  bc^- 

Next  to  monomials,  binomials  are  those,  which  are  the  least 
complicated.     We  begin  therefore  with  these. 

Below  are  several  of  the  first  powers  of  the  binomial  a:  -f-  «> 
viz. 

{x-^ay  =  x-\-a 

{x  +  aY  =  x'-{-2ax-{-a' 

{x-\'af  =  x'-{-2ax^-\-3a''x  +  a^ 

{x  +  ay  =  x*-\-  4:ax^  +  6aV  +  U^x  +  a\ 

We  have  formed  the  different  powers  of  x~\-am  this  table  by 
the  continued  multiplication  of  a:  -|-  ^  into  itself.  In  this  way 
we  arrive  only  at  particular  results.  To  form  any  of  the  higher 
powers,  the  process  of  multiplication  must  still  be  continued. 
This  would  be  tedious,  especially,  as  the  power  to  which  the 
binomial  is  to  be  raised,  becomes  more  and  more  elevated.  We 
proceed,  therefore,  to  investigate  a  method,  by  which  a  binomial 
may  be  raised  to  any  power  whatever,  without  the  necessity  of 
forming  the  inferior  powers.  This  method  was  discovered  by 
Newton.  The  principle  on  which  it  is  founded  is  called  the 
Binomial  Theorem.  The  most  simple  and  elementary  demon- 
stration of  this  theorem  depends  upon  the  theory  of  combinations, 
10  which  we  shall  first  attend. 

THEORY    OF    COMBINATIONS. 

125.  The  results,  obtained  by  writing  one  after  the  other,  in 
every  possible  way,  a  given  number  of  letters,  in  such  a  manner, 
that  all  the  letters  will  enter  into  each  result,  are  called  permu- 
tations. 

Let  there  be,  for  example,  two  letters,  a  and  b.  These  give,  it 
is  evident,  two  permutations,  ab^  ha. 


THEORY   OF   COMBINATIONS.  173 

Again,  let  there  be  three  letters,  «,  h  and  c.  If  we  set  apart 
one  of  the  letters,  a  for  example,  the  remaining  letters  give 
two  permutations,  viz.  3c,  ch;  placing  next  the  a  at  the  right 
of  each  of  these,  we  have  two  permutations  of  three  letters, 
viz.  hca^  cha;  but  each  of  the  remaining  letters  h  and  c,  being 
set  apart  in  the  same  manner,  will  also  furnish  each  two 
permutations  of  three  lettters ;  ivhence  the  permutations  of  three 
letters  will  be  equal  to  the  permutations  of  two  letters^  multiplied 
by  three. 

In  like  manner  the  permutations  of  four  letters  will  be  found 
equal  to  the  permutations  of  three  letters  multiplied  by  four. 

And  in  general,  the  permutations  of  any  number  whatever  n 
of  letters^  will  be  equals  it  is  evident,  to  the  permutations  of  n  —  1 
letters,  multiplied  by  n  the  number  of  letters  employed. 

Let  Q  represent  the  permutations  of  ?i  —  1  letters,  then  Qn 
will  represent  the  permutations  of  n  letters;  thus  Qn  will  be  a 
general  formula  for  permutations. 

In  the  general  formula  Q  n,  let  n  =  2,  then  Q  will  be  1 ; 
whence  1  X  2  will  be  the  permutations  of  two  letters.  Again 
let  71  =  3,  then  Q  will  be  1X2;  whence  1X^X3  will  be 
the  permutations  of  three  letters.  In  like  manner  the  permuta- 
tions of  4  letters  will  be  1  X  2  X  3  X  4.  The  following  rule 
for  permutations  will,  therefore,  be  readily  inferred,  viz.  Multi- 
ply in  order  the  natural  numbers,  1,  2,  3,  4,  ^c.  to  the  number 
denoting  the  letters  employed  inclusive;  the  result  ivill  be  the 
permutations  of  the  given  number  of  letters. 

126.  When  a  given  number  of  letters  are  disposed  in  order 
one  after  the  other  in  every  possible  way,  2  and  2,  3  and  3,  and, 
in  general,  n  and  ti  at  a  time,  the  number  of  letters  taken  at  a 
time  being  always  less  than  the  given  number  of  letters,  the 
results  obtained  are  called  arrangements, 

0* 


174  ELEMENTS    OF   ALGEBRA. 

Let  it  be  required  to  form  the  arrangements  of  three  letters, 
ay  b,  and  c,  two  and  two  at  a  time. 

a,  ab 
,  ac 

b,  ba 

be 

c,  ca 
cb 

Setting  apart  first  one  of  the  letters,  a  for  example,  we  write 
after  this  letter  each  one  of  the  reserved  letters  b  and  c ;  we 
thus  form  two  of  the  arrangements  sought,  viz.  ab,  ac ;  setting 
apart  next  the  letter  Z*,  and  writing  by  its  side  each  one  of  the 
reserved  letters  a  and  c,  we  form  two  more  of  the  arrangements 
sought,  viz.  hu,  he;  pursuing  the  same  course  with  the  re- 
maining letter  c,  we  have  in  the  result,  it  is  plain,  all  the  ar- 
rangements required  and  no  more ;  lohence  the  arrangements  of 
three  letters  2  and  2  at  a  time,  will  le  equal  to  the  arrangements 
of  the  same  letters  one  at  a  time,  multiplied  by  the  number  of 
letters  reserved. 

Let  it  be  required  next  to  form  the  arrangements  of  four  letters 
a,  b,  c,  d,  three  and  three  at  a  time. 

ab,         ab  c  c  a,         cab 

a bd  cad 


ac, 

acb 
acd 

cb, 
cd, 
da, 
db, 
dc, 

cb  a 
cbd 

ad, 

adb 
adc 

cda 
cdb 

ba. 

b  ac 
bad 

dab 
dac 

he, 

bca 
bed 

dba 
dbc 

bd, 

bda 
bdc 

dca 
deb 

Having  formed  the  arrangements  of  the  given  letters,  2  and 


THEORY    OF    COMBINATIONS.  176 

2  at  a  time,  we  set  apart  one  of  these,  ab  for  example,  and  write 
successively  by  its  side  each  one  of  the  reserved  letters  c  and  dy 
we  thus  form  two  of  the  arrangements  sought,  viz.  abc,  abd. 
The  same  being  done  with  each  one  of  the  remaining  arrange- 
ments of  the  given  letters,  2  and  2  at  a  time,  we  obtain,  it  is 
evident,  all  the  arrangements  required  and  no  more.  Thus,  the 
arrangements  of  4  letters  taken  3  and  ^  at  a  time,  ivill  be  equal 
to  the  arrangements  of  the  same  letters,  taken  2  and  2  at  a  timet 
multiplied  by  the  number  of  letters  reserved. 

In  like  manner,  understanding  by  letters  reserved  those  which 
remain,  when  the  given  letters  are  taken  one  less  than  the  re- 
quired number  at  a  time,  we  have  the  arrangements  of  any 
number  m  of  letters,  taken  n  arid  n  at  a  time,  equal  to  the 
arrangeinents  of  the  same  letters,  n  —  1  at  a  time,  multiplied  by 
the  number  of  letters  reserved. 

Let  P  represent  the  arrangements  of  m  letters  n  —  1  at  a 
time ;  it  being  required  to  take  the  letters  n  and  n  at  a  time,  the 
reserved  letters  will  be  77i — {n — 1),  or  m  —  7i-\-l;  thus  the 
arrangements  of  m  letters,  7i  and  ?i  at  a  time,  will  be  expressed 
by  the  formula, 

^{m  —  n+l). 

This  will  be  the  general  formula  for  arrangements. 
*    In  the  general  formula  'P  [m  —  n -\-  1),  let  n  equal  2.  '  In  this 
case  P  will  represent  the  arrangements  of  m  letters  1  at  a  time ; 
thus   P   will   equal   m;   whence    m  {m  —  1),  will    express   the 
arrangements  of  m  letters  2  and  2  at  a  time. 

Again,  in  the  general  formula  P  (?7^  —  n-^l),  let  n==3. 
In  this  case  P  will  represent  the  arrangements  of  ?ra  letters  2 
and  2  at  a  time;  P  will  therefore  equal  m  {m — 1) ;  whence 
m{?n — 1)  {m  —  2)  will  express  the  arrangements  of  m  letters 

3  and  3  at  a  time. 

In  like  manner  the  arrangements  of  m.  letters  4  and  4  at  a 
time,  will  be  expressed  hy  m  {m  —  I)  {m  —  2)  {m  —  3). 

From  inspection  of  the  above  formulas  the  following  rule  for 
arrangements  will  be  readily  inferred,  viz.     From  the  number 


176  ELEMENTS    OF    ALGEBRA. 

denoting  the  given  letters  subtract  successively  the  natural  num.* 
lers  1,  2,  3,  ^c.  to  the  number  which  denotes  the  letters  to  he 
taken  at  a  time  ;  multiply  these  several  remainders  and  the  num" 
her  denoting  the  given  letters  together ;  the  product  will  he  the 
arrangements  required. 

127.  Arrangements,  any  two  of  which  differ  at  least  by  one  of 
the  letters,  which  enter  into  them,  are  called  combinations. 

Let  it  he  proposed  to  determine  the  number  of  combinations  of 
three  letters,  a,  b,  and  c  taken  two  and  two  at  a  time. 

The  arrangements  of  these  letters,  two  and  two  at  a  time, 
will  be 

ah 
ha 

ac 
ca 

he 
ch 
Among  these  arrangements  we  have,  it  is  evident,  but  three 
combinations,  viz.  ab,  ac,  he,  each  one  of  which  is  repeated  as 
many  times  as  there  are  permutations  of  two  letters.  Hence  the 
combinations  of  three  letters  taken  2  and  2  at  a  time,  will  he  equal 
to  the  arrangements  of  three  letters  2  and  2  at  a  time,  divided  hy" 
the  permutations  of  two  letters. 

In  like  manner,  it  will  be  seen,  that  the  combinations  of  4 
letters,  3  and  3  at  a  time,  are  equal  to  the  arrangements  of  4 
letters,  3  and  3  at  a  time,  divided  by  the  permutations  of  three 
letters. 

And  in  general  the  combinations  of  m  letters,  n  and  n  at  a 
time,  will  be  equal,  it  is  evident,  to  the  arrangements  ofm.  letters^ 
n  and  n  at  a  time,  divided  by  the  permutations  of  n  letters. 

From  what  has  been  done,  we  have  therefore  the  following 
general  formula  for  combinations,  viz. 

^{m  —  n+iy 
Qn 


BINOMIAL   THEOREM.  177 

In  the  general  formula  let  w  =  2,  the  formula  which  results 

will  be  ^^^^. 

This  will  give  the  combinations  of  m  letters  2  and  2  at  a 
time. 

Again,  let  %  =  3 ;  the  formula  which  results  will  be 
m{7n  —  1)  {m  —  2) 

\7%7z        * 

This  will  give  the  combinations  of  m  letters  3  and  3  at  a  time. 

In   like    manner   we    obtain    — ^ ,  '       ^ — -r^ -.   a 

1.2.3.4 

formula  which  gives  the  combinations  of  m  letters  4  and  4  at 
a  time. 

From  inspection  of  the  formulas  obtained  by  making  %  =  2, 3, 4, 
&c.  in  the  general  expression,  we  may  infer  a  general  rule  for 
combinations  as  has  been  done  already  with  respect  to  permuta- 
tions and  arrangements. 

1.  For  how  many  days  can  7  persons  be  placed  in  a  different 
position  at  dinner  ? 

2.  How  many  words  can  be  made  with  5  letters  of  the  alpha- 
bet, it  being  admitted  that  a  number  of  consonants  may  make  a 
word? 

3.  How  many  combinations  can  be  made  of  24  letters  of  the 
alphabet  taken  two  and  two  at  a  time  ? 

4.  A  general  was  asked  by  his  king,  what  reward  he  should 
confer  on  him  for  his  services ;  the  general  only  desired  a  farthing 
for  every  file  of  ten  men  in  a  file,  which  he  could  make  with  a 
body  of  100  men.     At  this  rate  what  would  he  receive  ? 

BINOMIAL   THEOREM. 

128.  If  we  examine  with  attention  the  different  powers  of 
x-\-a^  art.  124,  it  will  be  easy  to  fix  upon  the  law,  according 
to  which  the  exponents  of  x  and  a  proceed.  But  it  will  not 
be  so  easy  to  determine  the  law  for  the  numerical  coefficients. 
y  12 


178 


ELEMENTS    OF   ALGEBEA. 


If  we  observe,  however,  the  manner  in  which  the  different 
terms  which  compose  a  power  are  formed,  we  shall  perceive 
that  the  numerical  coefficients  are  occasioned  by  the  reduction 
of  several  similar  terms  into  one,  and  that  these  similar  terms 
arise  from  the  equality  of  the  factors  which  compose  a  power. 
These  reductions,  it  is  easy  to  see,  will  not  take  place,  if  the 
second  terms  of  the  binomials  are  different.  AVe  begin  there- 
fore by  investigating  a  law  for  the  formation  of  the  product  of 
any  number  of  binomials  x-\-a,  x~\-b,  x-\-c.  .  .  ,  the  first 
terms  of  which  are  the  same  in  each,  while  the  second  are 
different. 


{x  +  a){x  +  b) 


x^  -\-a 
b 


+  a5 


{x-\-h)  (x  +  b)  {x-{-c)=x^  +  a 

b 


x  +  a){x  +  b){x  +  c){x  +  d) 


ac 
be 


=  x'-\-a 

x'  +  ab 

x'-^abc 

b 

ac 

abd 

c 

ad 

acd 

d 

be 

bed 

bd 

cd 

x-\-abe 


x-\-abcd. 


From  inspection  of  the  above  products,  which  we  have  formed 
by  the  common  rules  of  multiplication,  it  will  be  observed, 

1°.  hi  each  product  there  is  one  term  more  than  there  are  units 
in  the  number  of  factors. 

2°.  The  exponent  of  ^  in  the  first  term  is  the  same  as  the  num*- 
ber  of  factors,  and  goes  on  decreasing  by  unity  in  each  of  the  foU 
lowing  terms. 

3°.  The  coefficient  of  the  first  term  is  unity.  The  coefficient 
of  the  second  term  is  equal  to  the  sum  of  the  second  terms  of  the 
binomials  ;  that  of  the  third  term  is  equal  to  the  sum  of  the  differ- 
ent  combinations  or  products  of  the  second  terms  of  the  binomials 
taken  two  and  two  ;  that  of  the  fourth  is  equal  to  the  sum  of  the 
products  of  the  second  terms  of  the  binomials  taken  three  and, 


BINOMIAL   THEOREM.  179»' 

three,  and  so  on.  The  last  term  is  equal  to  the  product  of  the 
second  terms  of  the  binomials. 

129.  A\re  readily  infer  from  analogy,  that  the  same  law  wilF 
obtain,  whatever  be  the  number  of  factors  employed.  This  law 
may,  however,  readily  be  shown  to  be  general.  In  order  to  this, 
it  will  be  sufficient  to  show,  that  if  the  Jaw  be  true  for  the 
product  of  any  number  m  of  binomials,  it  will  also  be  true  for  the 
product  of  772  -|~  1  binomials. 

The  number  of  binomial  factors  being  represented  by  w, 
the  different  powers  of  x  will  be  a;*",  a:*""*,  a;'""^  &c.  Let 
A,  B,  C,  .  .  .  U  denote  the  quantities,  by  which  these  powers 
beginning  with  z'""^  are  to  be  multiplied;  but  as  the  number 
of  terms  must  remain  indeterminate,  until  m  receives  a  particular 
value,  we  can  write  only  a  few  of  the  fifst  and  last  terms  of  the 
expression,  designating  the  intermediate  terms  by  a  series  of 
points. 

The  product  of  any  number  m  of  factors  will  then  be  repre- 
sented by  the  expression 

a:'"  +  Aa;'"-^  +  Bx"'--  4   Ca:'"-^  .  .  U. 

Multiplying  this  expression  by  a  new  factor  a:  -[-  K,  it  becomes 

2:—=^..,.  UK. 


+  ^  +  A 
K 


a:'"  +  B 
AK 


BK 


Here  the  law  for  the  exponents  is  evidently  the  same,  as  in 
the  first  expression.  With  respect  to  the  coefficients,  it  is 
evident,  1°.  that  the  coefficient  of  the  first  term  is  unity.  2°* 
A  -(-  K,  the  coefficient  of  the  second  term,  is  equal  to  the  sum 
of  the  second  terms  of  the  w,  -(-  1  binomials.  3°.  Since  B  by 
hypothesis  expresses  the  sum  of  the  second  terms  of  the  m 
binomials  taken  two  and  two,  and  AK  expresses  the  sum  of  the 
second  terms  of  the  m  binomials  multiplied  each  by  the  new 
second  term  K,  B  -f-  AK,  the  coefficient  of  the  third  term,  will 
be  the  sum  of  the  products  two  and  two  of  the  second  terms  of 
the  m-\'\  binomials. 

In  the  same  manner  C  -|-  BK,  it  is  easy  to  see,  will  be  tlw 


180 


ELEMENTS    OF    ALGEBRA. 


sum  of  the  products  three  and  three  of  the  second  terms  of  the 
m-\-l  hinomials,  and  so  on.  4°.  The  last  term  UK  it  is  evi- 
dent, is  the  product  of  the  m-\-  1  second  terms. 

The  law  laid  down,  art.  128,  being  true  therefore  for  expres- 
sions of  the  fourth  degree  will,  from  what  has  just  been  de- 
monstrated, be  true  for  those  of  the  fifth;  and  being  true  for 
expressions  of  the  fifth  degree,  it  will  be  true  for  those  of  the 
sixth  and  so  on ;  thus  it  is  general. 

130.  If  in  the  different  products  which  have  been  formed, 
art.  128,  we  make  b,  c  and  d  each  equal  to  a,  these  products  will 
be  converted  into  powers  of  x  -\-  a,  thus 


{x  +  a)  {x-{-b)  =  {x-\-af=^x'-{-a 


a 


x'  +  a' 


=  {x  +  a] 

' 

a^  +  a' 

x'  +  a' 

a' 

a^ 

a' 

a' 

a' 

(^ 

a' 

a' 

+^- 


x  +  a'. 


{x-\-a){x-]-b)  {x-\-c)  =  {x-{-aY  =  a^-}-a 

a 
a 

x-\-d){x-\-b){x^  c)  {x  +  d): 

=  x'  +  ^ 
a 
a 
a 


Comparing  these  expressions  with  the  different  products, 
from  which  they  are  derived,  we  perceive  1°.  that  the  multi- 
plier of  X  in  the  second  term  has  been  converted  into  the  first 
power  of  a,  repeated  as  many  times  as  there  are  units  in  the 
number  of  binomials  employed,  or  which  is  the  same  thing, 
as  there  are  units  in  the  exponent  of  x  in  the  first  term.  2°. 
That  the  multiplier  of  the  third  term  has  been  converted  into 
the  second  power  of  c,  repeated  as  many  times,  as  there  can 
be  formed  different  products  from  a  number  of  letters  equal  to 
the  number  of  binomials  employed,  taken  two  and  two  at  a 
time.  3**.  That  the  multiplier  of  the  fourth  term  has  been  con- 
verted into  the  third  power  of  a,  repeated  as  many  times  as 


BINOMIAL    THEOREM.  181 

there  can  be  formed  different  products  from  a  number  of  letters, 
equal  to  the  number  of  binomials  employed,  taken  three  and 
three  at  a  time,  and  so  on. 

131.  From  what  has  been  done,  it  is  evident,  therefore,  that 
whatever  be  the  power  to  which  a  binomial  a:  -|"  ^  is  to  be  raised, 
1°.  the  exponent  of  x  in  the  first  term  will  be  equal  to  the  expo- 
nent of  the  power,  and  that  it  will  go  on  decreasing  by  unity  in 
each  of  the  following  terms  to  the  last,  in  which  it  will  be  0. 
2°.  That  the  exponent  of  a  in  the  first  term  will  be  0,  in  the 
second  unity,  and  that  it  will  go  on  increasing  by  unity,  until  it 
becomes  equal  to  the  exponent  of  the  power  to  be  formed. 
3°.  The  numerical  coefficient  of  x  in  the  first  term  will  be  unity, 
in  the  second  it  will  be  equal  to  the  exponent  of  x  in  the  first 
term,  in  the  third  it  will  be  equal  to  the  number  of  products, 
which  may  be  formed  from  a  number  of  letters,  equal  to  the 
exponent  of  x  in  the  first  term,  taken  two  and  two  at  a  time,  in 
the  fourth  it  will  be  equal  to  the  number  of  products,  which  may 
be  formed  from  the  same  number  of  letters,  taken  three  and  three 
at  a  time  and  so  on. 

Let  it  be  required  to  form  the  5th  power  Qi  x-\-  a.  The  dif- 
ferent terms,  without  the  numerical  coefficients,  will  be  by  the 
preceding  rule  x"  -\- ax^  -\- c^ :»?  -\- c^ :i^  •\-  a^  x -\-  a^. 

With  respect  to  the  numerical  coefficients,  that  of  the  first 
term  will  be  1,  that  of  the  second  will  be  5,  that  of  the  third  will 
be  equal  to  the  number  of  products,  which  may  be  formed  of  5 
letters  taken  2  and  2,  that  of  the  fourth  will  be  equal  to  the  num- 
ber of  products,  which  may  be  formed  of  5  letters  taken  3  and  3, 
and  so  on.     Thus  the  numerical  coefficients  will  be 

1,  5,  10,  10,  5,  1 ; 
whence 

{x  +  af=^7?^5ax'  +  lOa^x'  +  lOa^r'-f-  5a'x  +  a'. 

Let  it  next  be  required  to  raise  x-\-  aio  the  vTzth  power,  we 
shall  have,  according  to  the  preceding  rule,  for  a  few  of  the  first 
terms  without  the  numerical  coefficients 

x"^  -\-  ax"^-'  -{-  a^x"^-^  -\-  a^x"^-^  -\- .  ., 
p 


182  ELEMENTS    OF    ALGEBRA. 

.     Here  the  numerical  coefficients  cannot  be  determined  until  we 

assign  a  particular  value  to  m  ;  by  the  preceding  rule,  however, 

the  numerical  coefficient  of  the  second  term  will  be  equal  to  wz, 

whatever  the  value  of  m  may  be.     In  the  development  therefore 

of  [x  -|-  cl)""  we  write  m  for  the  coefficient  of  the  second  term. 

With   respect  to  the   third  term   the   numerical  coefficient  will 

be  equal  to  the  number  of  products,  which  may  be  formed  of 

m  letters  2  and  2  at  a  time ;  this  is  expressed  by  the  formula 

m{vi  —  1)  .      ,        -       mim — 1)  .      ,  zr  •     .    r.-u 

— - — we  write  theretore  — - — - —  lor  the  coemcient  oi  the 

...                „             ...                   m{7n  —  \)  [m  —  2)      .„   , 
third  term,     bor  a  similar  reason  — ^ = — s^^i ^^"^^  "® 

the  coefficient  of  the  fourth  term,  and  so  on.     We  have  then 

....  a'". 

From  inspection  of  the  different  terms  of  this  development 
it  will  be  perceived,  that  the  coefficient  of  the  fourth,  for  example, 

is  formed  by  multiplying — ^r— ?  the  coefficient  of  the  third 

1    .   o 

term,  by  7n  —  2  the  exponent  of  x  in  this  term,  and  dividing 
by  3  the  number,  which  marks  the  place  of  this  term.  It 
will  be  perceived,  also,  that  the  coefficient  of  the  third  term  is 
formed  in  the  same  manner  by  means  of  the  second  term,  and 
that  of  the  second  by  means  of  the  first.  We  readily  infer, 
therefore,  the  following  rule,  by  which  to  form  the  coefficient 
of  any  term  v/hatever,  viz.  Multiply  the  coefficient  of  the  pre' 
ceding  term  by  the  exponent  of  x  in  that  term,  arid  divide  the 
product  by  the  number  which  marks  the  place  of  that  term  from 
the  left. 

From  what  has  been  done,  we  have  therefore  the  following 
rule,  by  which  to  raise  a  binomial  to  any  power  whatever,  viz. 
P.  The  coefficient  of  x  in  the  first  term  is  unity,  and  its  exponent 
is  equal  to  the  number  of  units  in  the  degree  of  the  power  to  which 
the  binomial  is  to  be  raised.     2°.   To  pass  from  any  term  to  the 


BINOMIAL   THEOREM.  1S3 

folloioing,  we  multiply  the  numerical  coefficient  by  the  exponent 
of  X  in  that  term^  dioide  by  the  number  which  marks  the  place  of 
that  term  from  the  left,  increase  by  unity  the  exponent  of  a  and 
diminish  by  unity  the  exponent  of  x. 

According  to  this  rule 
{x-\-af=^x'  +  Qax'+\5d'x'  +  2Qa^x^-]r^5a'x'-\-Qa'x-{-a\ 

132.  It  sometimes  happens,  that  the  terms  of  the  proposed 
binomial  are  affected  with  coefficients  and  exponents.  The 
following  example  will  exhibit  the  course  to  be  pursued  in  cases 
of  this  kind. 

Let  it  be  proposed  to  raise  the  binomial  ^c^b  —  3aic  to  the 
fourth  power. 

Putting  ^a^b  =  x,  and  — 2abc  =  y,  we  have 

{X  +  yY  =  x'  +  ^x'y  +  Qx'f  +  ^xf -\-  y\ 
Substituting  next  for  x  and  y  their  values,  we  have 

(4^2^  _  ^abcY  =  {^aHf  +  4  {^aHf  (—  3aic)  +  . .. . 
6  {^aHf  {—^abcf^^  (4a^i)  {  —  ^abcf  +  {—^abc)\ 
or  performing  the  operations  indicated,  we  have 

{^aH  —  ^abcy=25QaH'  —  ima'b'c-l-SUa%*(?.  . 
—  ^^2a'b'c^-{-Sla'b'c\   ■ 
The   terms    produced    by   this   development    are   alternately- 
positive  and  negative.     This,  it  is  evident,  should   always  be 
the  case,  when  the  second  term  of  the  proposed  binomial  has  the 
sign  — . 

133.  The  powers  of  any  polynomial  whatever  may  be  found 
by  the  binomial  theorem.  Let  it  be  proposed  to  find  for  example, 
the  third  power  of  the  trinomial  a-^b  '\-  c. 

In  order  to  apply  the  rule  to  this  case,  we  put  a-\-b=:m; 
the  proposed  is  then  reduced  to  the  binomial  m  -\-  c,  and  we  have 

{m  +  cf  =  771^  +  3';^^'c  -\-'^me -\- & 
whence,  restoring  the  value  of  m,  we  have 

(a  +  ^  +  cf  ==  fl3  +  3a=^  4-  3^^=^  +  b^ 

+  3a=c  +  6a^c  +  33'^c 
+  3ac2  +  33c^ 
+  c3. 


184  ELEMENTS    OF   ALGEBRA. 

The  same  process,  it  is  easy  to  see,  may  be  applied  to  any 
polynomial  whatever. 

MISCELLANEOUS    EXAMPLES. 

1.  To  find  the  third  power  of  2  a  —  ^  -[-  c^ 

2.  To  find  the  seventh  power  of  Sa^  —  2  a*. 

3.  To  find  the  fifth  power  oi  a'  —  c  —  2d. 

4.  To  find  the  third  power  of  2^^  —  4a^  +  S^*^. 


SECTION  XVII.— Roots  of  Compound  Quantities. 

134.  We  pass  next  to  the  extraction  of  the  roots  of  com- 
pound quantities,  beginning  with  the  third  or  cube  root  of  num- 
bers. 

In  the  following  table,  we  have  the  nine  first  numbers,  with 
their  third  powers  or  cubes  written  under  them  respectively. 
1,      2,       3,       4,       5,       6,       7,       8,       9, 
1,      8,    27,    64,  125,  216,  343,  512,  729. 

By  inspection  of  this  table,  it  will  be  perceived,  that  among 
numbers  consisting  of  two  or  three  figures,  there  are  nine  only, 
which  are  perfect  third  powers,  the  others  have  each  for  a  root 
an  entire  number  plus  a  fraction. 

If  the  proposed  number  consists  of  not  more  than  three  figures, 
its  third  root  or  that  of  the  greatest  third  power  contained  in  it, 
may  be  found  immediately  by  the  above  table. 

Let  it  be  proposed  to  extract  the  third  root  of  a  number,  con- 
sisting of  more  than  three  figures,  103823,  for  example. 

The  proposed  being  comprised  between  1000,  the  third  power 
of  10,  and  1000000,  the  third  power  of  100,  its  root  will 
consist  of  two  places,  units  and  tens.  To  return  therefore  from 
the  proposed  to  its  root,  let  us  observe  the  manner,  in  which  the 


ROOTS   OF   COMPOUND   QUANTITIES.  185 

units  and  tens  of  a  number  are  employed  in  forming  the  third 
power  of  this  number.  For  tliis  purpose  designating  the  tens  by 
a  and  the  units  by  3,  we  have 

From  this  we  learn,  that  the  third  power  of  a  number  consisting 
of  units  and  tens,  is  composed  of  the  third  poioe?'  of  the  tens,  the 
triple  product  of  the  square  of  the  tens  by  the  units,  the  triple 
product  of  the  tens  by  the  square  of  the  units,  a?id  the  third  power 
of  the  units. 

If  then  we  CE^n  determine  in  the  proposed  the  third  power  of 
the  tens,  the  tens  of  the  root  will  be  found  by  extracting  the  third 
root  of  this  part.  The  third  power  of  the  tens,  it  is  evident,  can 
have  no  significant  figure  below  the  fourth  place,  the  three 
figures  on  the  right  will,  therefore,  form  no  part  of  the  third 
power  of  the  tens,  and  may  on  this  account  be  separated  from 
the  rest  by  a  comma.  The  third  power  of  the  tens  will  then  be 
contained  in  103,  the  part  at  the  left  of  the  comma.  The  great- 
est third  power  contained  in  103  is  64,  the  root  of  which  is  4;  4 
is,  therefore,  the  significant  figure  in  the  tens  of  the  root  sought. 
Indeed,  the  proposed  is  evidently  comprised  between  64000,  the 
third  power  of  40  or  4  tens,  and  125000  the  third  power  of  50 
or  5  tens.  The  root  sought  is,  therefore,  composed  of  4  tens  and 
a  certain  number  of  units  less  than  ten. 

The  tens  of  the  root  being  thus  obtained,  we  subtract  the  third 
power  64  from  103,  the  part  of  the  proposed  at  the  left  of  the 
comma,  and  to  the  remainder  bring  down  the  figures  at  the  right. 
The  result  of  this  operation,  39823,  must  contain,  from  what  has 
been  said,  the  triple  product  of  the  square  of  the  tens  by  the 
units,  together  with  the  two  remaining  parts  in  the  third  power 
of  the  root  sought. 

The  square  of  the  tens,  it  is  evident,  will  contain  no  signifi- 
cant figure  less  than  hundreds ;  on  this  account  we  separate  23, 
the  two  figures  on  the  right  of  the  remainder  39823,  from  the 
rest  by  a  comma;  398,  the  figures  on  the  left  of  the  comma,  will 
then  contain  the  triple  product  of  the  square  of  the  tens  of  the 


186  ELEMENTS    OF    ALGEBRA. 

root  sought  by  the  units  and  something  more,  in  consequence  of 
the  hundreds  arising  from  the  two  remaining  parts  of  the  third 
power  of  the  root  sought.  Dividing  therefore  398  by  48,  the 
triple  product  of  the  square  of  the  tens,  already  found,  the 
quotient  8  will  be  the  unit  figure  sought,  or,  from  what  has  been 
said,  it  may  be  too  large  by  1  or  2. 

To  determine  whether  8  be  the  right  unit  figure  we  raise  48  to 
the  third  power.  This  gives  110592,  a  number  greater  than  the 
proposed ;  8  is,  therefore,  too  large  for  the  unit  figure.  We 
next  try  7 ;  47  raised  to  the  third  power  gives  108823.  The 
proposed  is,  therefore,  a  perfect  third  power,  the  root  of  which 
is  47. 

The  operation  may  be  exhibited  as  follows. 

103,823  I  47 
64 


398,23  I  48 


103,823 


Any  number  however  large  may  be  considered  as  composed 
of  units  and  tens ;  the  process  for  finding  the  root  may  therefore 
be  reduced  to  that  of  the  preceding  example. 

Let  it  be  proposed,  for  example,  to  find  the  third  root  of. 
43725678.  Considering  the  root  of  this  number  as  composed 
of  units  and  tens,  678  the  three  right  hand  figures,  it  is  evident, 
will  form  no  part  of  the  third .  power  of  the  tens.  On  this  ac- 
count we  separate  them  from  the  rest  by  a  comma.  The  third 
power  of  the  tens  being  contained,  then,  in  the  part  at  the  left 
of  the  comma,  we  obtain  the  tens  of  the  root  sought  by  ex- 
tracting the  third  root  of  this  part.  Considering  therefore, 
for  the  moment,  the  part  of  the  proposed  43725  as  a  separate 
number,  its  third  root,  it  is  evident,  may  be  found  as  in  the 
preceding  example.  Performing  the  operations,  we  have  35 
for  the  root  and  a  remainder  of  850.     There  will  therefore  be 


ROOTS    OF    COMPOUND    QUANTITIES.  187 

3«)  tens  in  the  root  of  the  proposed,  and  in  order  to  find  the  units, 
we  bring-  down  the  three  right  hand  figures  678  by  the  side  of 
850,  which  gives  850678.  Separating  next  the  two  right  hand 
figures  of  this  last  from  the  rest  by  a  comma,  and  dividing  the 
part  on  the  left  by  the  triple  square  of  the  tens  already  found, 
we  obtain  2  for  the  unit  figure  of  the  root  sought.  To  determine 
whether  this  is  the  right  figure,  we  raise  352  to  the  third  power, 
which  gives  43614208,  a  result  less  than  the  proposed.  352  is, 
therefore,  the  root  of  the- proposed  to  within  less  than  a  unit. 
The  operation  may  be  exhibited  as  follows: 


43,725,676 
27 


352 


Is  Dividend  167,25  [  27         1st  Divisor 

Third  power  of  35     42875 


2d  Dividend  8506,78  |  3675,        2d  Divisor 

Third  power  of  352    43614208 

Remainder  111470 

The  same  process,  it  is  easy  to  see,  may  be  extended  to  any 
number  however  large.  The  rule,  therefore,  for  the  extraction 
of  the  third  root  will  be  readily  inferred. 

If  it  happens,  that  the  divisor  is  not  contamed  in  the  dividend 
prepared  as  above,  a  zero  must  be  placed  in  the  root,  and  the 
next  figure  brought  down  to  form  the  dividend. 

EXAMPLES. 

1.  Find  the  third  root  of  150568768.  Ans.  532. 

2.  Find  the  third  root  of  205483447701.  Ans.  5901. 

3.  Find  the  third  root  of  32977340218432.        Ans.  32068. 
135.  If  the  proposed  be  a  fraction  its  third  root  is  found  by 

extractinf?  the  third  root  of  the  numerator  and  denominator. 


lus  y/ , 


Thus  I  /  |,is|: 


188  ELEMENTS    OF   ALGEBRA. 

If  the  denominator  is  not  a  perfect  third  power,  it  may  be 

made  so,  by  multiplying  both  terms  by  the  square  of  the  denom- 

3 
inator ;  thus  if  the  proposed  be  -,  we  multiply  both  terms  by  49 ; 

147 

the   fraction   then  becomes  ^-r^,  the   root  of  which  is  nearest 

t>4«3 

-,  accurate  to  within  less  than  -. 
7  7 

136.  We  have  seen,  art.  94,  that  the  square  root  of  an  entire 
number,  which  is  not  a  perfect  square,  cannot  be  exactly  assigned. 
The  same  is  true  with  respect  to  the  roots  of  all  entire  numbers, 
which  are  not  perfect  powers  of  a  degree  denoted  by  the  index 
of  the  root. 

The  third  root  of  a  number  which  is  not  a  perfect  third  power 

may  be  approximated  by  converting  the  number  into  a  fraction, 

the  denominator  of  which  is  a  perfect  third  power.     Thus  let  it 

be  required  to  find  the  approximate  third  root  of  15.     This  num- 

1      r         15  X  12'      25920     ,       ^.  J 
ber  may  be  put  under  the  form  — —3 —  ==———,  the  third 

29  5  1 

root  of  which  is  j^ ,  or  2  — ,  accurate  to  within  less  than  — .     If 

a  greater  degree  of  accuracy  were  required,  we  should  convert 
the  proposed  into  a  fraction,  the  denominator  of  which  is  the 
third  power  of  some  number  greater  than  12. 

In  such  cases  it  is  most  convenient  to  convert  the  proposed 
number  into  a  fraction,  the  denominator  of  which  shall  be  the 
third  power  of  10,  100,  1000,  &c. 

Thus  if  it  be  required  to  find  the  third  root  of  25  to  within 
.001,  we  convert  the  proposed  into  a  decimal,  the  denominator 
of  which  is  the  third  power  of  1000,  viz.  25.00Q000000,  the 
third  root  of  which  is  2.920  to  within  .001 ;  we  have  then 
>v/25=:  2.920,  accurate  to  within  less  than  .001. 

To  approximate  therefore  the  third  root  of  an  entire  number  by 
means  of  decimals,  we  annex  to  the  'proposed  three  times  as  many 
zeros  as  there  are  decimal  places  required  in  the  rout,  we  then 


Hoots  of  compound  quantities.  189 

extract  the  root  of  the  number  thus  prepared  to  within  a  unit, 
and  point  off  for  decimals,  as  mamj  places  as  there  are  decimal 
figures  required  in  the  root. 

187.  If  'the  proposed  number  contain  decimals,  beginning  at 
the  place  of  units,  we  separate  the  number  both  to  the  right  and 
left  into  periods  of  three  figures  each,  annexing  zeros,  if  neces- 
sary, to  complete  the  right  hand  period  in  the  decimal  part.  We 
then  extract  the  root,  and  point  off  for  decimals  in  the  root 
as  many  places  as  there  are  periods  in  the  decimal  part  of  the 
power. 

If  the  proposed  be  a  vulgar  fraction,  the  most  simple  method 
of  finding  the  third  root  is  to  convert  the  proposed  into  a  decimal, 
the  number  of  places  in  which  shall  be  equal  to  three  times  the 
number  of  decimal  figiires  required  in  the  root.  The  question  is 
thus  reduced  to  extract  the  third  root  of  a  decimal  fraction. 

EXAMPLES. 

1.  Fnd  the  approximate  third  root  of  79.  Ans.  4.2908. 

2.  Find  the  approximate  third  root  of  ^J.  Ans.  0.824. 

3.  Find  the  approximate  third  root  of  3.00415. 

Ans.  1.4429. 

4.  Find  the  approximate  third  root  of  15f .  Ans.  2.502. 
138.  By  processes  altogether  similar  to  that,  which  we  have 

employed  in  tlm  extraction  of  the  third  root  of  numbers,  we 
may  extract  the  root  of  any  degree  whatever.  The  method  of 
extracting  the  root  of  any  degree  whatever,  in  the  case  of  alge- 
braic quantities,  is  also  founded  upon  the  same  principles.  The 
following  example  will  be  sufficient  to  illustrate  the  course 
to  be  pursued,  whatever  the  degree  of  the  root  required  may  be. 

Let  it  be  proposed  to  extract  the  fifth  root  of  the  polynomial 
32a''  —  80aH'  +  S0an'^^0a*b'-\-l0a'b'^  —  b'^. 

The  proposed  being  arranged  with  reference  to  the  powers 
of  the  letter  a,  we  seek  the  fifth  root  of  the  first  term  32  a" 


19(^  ELEMENTS    OF    ALGEBRA. 

Its  root  2  a'  will  be  the  first  term  of  the  root  sought.  "We 
write,  therefore,  2d^  in  the  place  of  the  quotient  in  division,  and 
subtracting  its  fifth  power  from  the  whole  quantity,  we  have  for 
a  remainder 

The  second  term  of  the  binomial  [a  -|-  if  is  5  a^  3  ;  this  shows, 
that  in  order  to  obtain  the  second  term  of  the  root,  we  must 
divide  — S^a^b^,  the  second  term  of  the  proposed,  by  five  times 
the  4th  power  of  ^c^,  the  term  of  the  root  already  found.  Per- 
forming the  operation  we  obtain  —  b^.  This  will  be,  therefore, 
the  second  term  of  the  root.  Raising  2c^  —  b^  to  the  fifth  power, 
it  produces  the  quantity  proposed.  The  root  is  therefore  obtained 
exactly.  If  the  root  contained  more  than  two  terms,  it  would  be 
necessary  to  subtract  the  fifih  power  of  2a^  —  b^  from  the  pro- 
posed quantity,  and  then  in  order  to  find  the  next  term  of  the 
root,  to  divide  the  first  term  of  the  remainder  by  five  times  the 
4th  power  of  2a^  —  b^.  In  this  case,  however,  only  the  first  term 
of  the  divisor  would  be  used ;  we  should  have  therefore  the  same 
divisor,  that  was  used  the  first  time. 

139.  When  the  index  of  the  root  has  divisors  the  root  may  be 
found  more  readily  than  by  the  general  method.  Thus  the 
fourth  root  may  be  found  by  extracting  the  square  root  twice 
successively ;  for  the  square  root  of  a^  is  a^^  and  that  of  c^  is  a, 
the  fourth  root  of  a*.  In  general,  all  roots  of  a  degree  marked 
by  4,  8  or  any  power  of  2  may  be  found  by  si^cessive  extrac- 
tions of  the  square  root.  Roots,  the  indices  oT  which  are  not 
prime  numbers,  may  be  reduced  to  others  of  a  degree  less 
elevated.  The  6th  root,  for  example,  may  be  found  by  first 
extracting  the  square  and  then  the  third  root ;  for  the  square  root 
of  c^  is  a^  and  the  third  root  of  a^  is  a, 

EXAMPLES. 

1.  To  find  the  third  root  of  Qs^  +  ZQj?  +  54a:  +  27. 

Ans.  22: +  3. 


CALCULUS    OF    RADICAL    EXPRESSIONS.  191 

2.  To  find  the  third  root  of  x' +  6x' —  4.0x' +  96x  —  64 

Ans.  3^-\-2x  —  4. 

3.  To  find  the  third  root  oi  x'  —  Qa^-{-  \5x'  —  20a:^  +  \5a? 
^Qx-\   \.  Ans.  x'  —  2x-\-\. 

4.  To  find  the  third  root  oi  21  x^  —  5^x^  -\-Q'^x'  ^^^x^  + 
21.£2  — e^r+l.  Ans.  ^x'  —  2x-\-\, 

5.  To   find   the   fourth   root   of   16  a*  —  96  a?  x -{- 216  a"  x" — 
\n6ax'^-^8lx*.  Ans.  2a— 3a:. 


SECTION  XVIII. — Calculus  of  Radical  Expressions. 

140.  Radical  expressions,  the  roots  of  which  cannot  be  found 
exactly,  frequently  occur  in  the  solution  of  questions.  On  this 
account  mathematicians  have  been  led  to  investigate  rules  for 
performing  upon  quantities  subjected  to  the  radical  sign,  the 
operations  designed  to  be  performed  upon  their  roots.  In  this 
way  the  calculations  required  in  the  solution  of  a  question  are 
frequently  rendered  more  simple,  and  the  extraction  of  the  root 
is  left  to  be  performed  at  last,  when  the  radical  expression  is 
reduced  to  the  most  simple  form,  which  the  nature  of  the  question 
will  allow. 

^DDITION    AND    SUBTRACTION. 

141.  Radical  expressions  of  the  same  degree,  and  which  have 
the  quantities  placed  under  the  radical  sign  also  the  same,  are 
said  to  be  similar. 

The  addition  and  subtraction  of  similar  radicals  is  performed 
upon  the  coefiicients.     Thus  the  sum  of  the  radicals 
34/*,  94/3,  is  12^3;  the  sum  o(  a4/W,  b4/¥7,—cA/l^ 
is  {a  +  b  —  c)4/¥7. 

In  like  manner   9\/a*c,  subtracted  from    12  \^a^c  gives 


192  ELEMENTS    OF    ALGEBRA. 


Vc,  and  bwab^  subtracted  from  a^/ab^  gives 

Radical  expressions,  which  are  at  first  dissimilar,  frequently 
become  similar  when  reduced  to  their  most  simple  form.  Thus, 
let  it  be  required  to  add  5 \/2a?b^  and  a  s/'E\cFW.  These  ex- 
pressions, reduced  to  their  most  simple  form,  become  5aw2c^b^, 
^a^2a^b'^;  their  sum  is  therefore 

Sa^/2^. 

The  addition  and  subtraction  of  dissimilar  radicals  can  be 
effected  only  by  means  of  the  signs  -\-  and  — . 

EXAMPLES. 

1.  To  find  the  sum  of  ^a'f^bl,  and  a^W^b^. 

2.  To  find  the  sum  of  ac?  s/T^Wc,  and  3  Va^^V. 

3.  To  find  the  sum  of  ^/^WJb?,  and  ^^sflE^Tc. 

4.  To  find  the  sum  of  2  V8,  —  7  Vl8,  5^72,  and  —  V^O. 

Ans.  8V2. 

5.  To  find  the  sum  of  8  Vi  —J'^IS,  4  a/27,  and  — 2  Vi^g- 

Ans.  VV3. 

6.  To  find  the  sum  of  2^1^  V^O,  —  Vl5,  and  Vf- 

Ans.  ffVlSr 

7.  To  find  the  sum  of  ^/lQa'b\  and  VdOa'bK 

Ans.  {3aH-{-5ab)^2ab, 

8.  To  subtract  sT^^Wc  from  1  as/¥c. 

9.  To  subtract  ^24  from  \/l92.  Ans.  24/3. 

10.  To  subtract  l/^^  from  l]/^?^. 

Ans,  (3a- 1)^^. 


CALCULUS   OF   RADICAL   EXPRESSIONS.  199 


MULTIPLICATION   AND   DIVISION. 


142.  Let  it  be  required  to  multiply  /^  a  by  iHJh^  we  have 
/i/ay^^b  =  ^ab;  for  ^/fl  X  \/^  raised  to  the  seventh 
power  gives  ab  for  the  resuh,  and  wab  raised  to  the  seventh 
power  gives  also  ab  for  the  result;  whence  the  seventh  powers 
of  these  expressions  being  equal,  the  expressions  themselves 
must  be  equal. 

The  same  reasoning  may  be  applied  to  all  similar  cases ;  we 
have,  therefore,  the  following  rule  for  the  multiplication  of  radi- 
cal expressions  of  the  same  degree,  viz.  Take  the  product  of  the 
quantities  under  the  radical  sign,  observing  to  place  the  result 
under  a  sign  of  the  same  degree. 

Let   it  next   be   required   to   divide   /^a  hy  A^b.     In  this 

,  /^a       J    /a      ^  j^a 

case    we    have    -57-7  =  1   /     - ;    lor    the    expressions    -^77, 
A/ o        Y        ^  w^ 

\X   -  being  raised  to  the  fifth  power  give   each  -\   these 

expressions  are  therefore  equal. 

We  have  then  the  following  rule  for  the  division  of  one  radical 
quantity  by  another  of  the  same  degree,  viz.  Take  the  quotient 
arising  from  the  division  of  the  quantities  under  the  radical 
sign,  recollecting  to  place  it  under  a  sign  of  the  sam£  degree. 


• 


EXAMPLES. 


1.  Multiply  4/4,  74/6,  and  ^4/5  together. 

Ans.  l^/^^. 

2.  Multiply  5^3,  7;^  f,  and  a/ 2  together.         Ans.  140. 

3.  Multiply7  +  2V6by9  — 5V6.      Ans.  3—17^6. 

4.  Multiply  8  +  2V7by8  —  2V7.  Ans.  36. 

5.  Multiply5V3  — 7V6by2V8  — 3. 

Ans.  41V6  — 71V3. 

13  n^  ^ 


194  ELEMENTS    OF   ALGEBRA. 

6.  Divide  V243  by  VlS.  Ans.  SJ. 

7.  Divide  /v/24^^  by  ^8TJb.  Ans.  |a^3. 

8.  Divide  1  by  r-|-^2.  Expressing  the  quotient  in  the 
form  of  a  fraction,  and  multiplying  both  terms  by  1  —  >^2 
we  have  Ans.  ^2  —  1. 

9.  Divide  l  +  V6by2V2  — V-*^-      Ans.  a/^+V^. 

FORMATION   OF   POWERS   AND   EXTRACTION   OF   ROOTS. 

143.  Let  it  be  required  to  raise  the  radical  ^a^b  to  the  third 
power ;  we  have 

{\/^f  =  ^^b  X  /^^  X  ^^  =  4/^3 
according  to  the  rule  established  for  multiplication. 

Whence  to  raise  a  radical  quantity  to  any  power ;  we  raise  ike 
quantity  placed  under  the  radical  sign  to  the  power  required^ 
observing  to  place  the  result  under  the  same  radical  sign. 

When  the  index  of  the  radical  is  a  multiple  of  the  exponent 
of  the  power  to  which  the  radical  is  to  be  raised,  it  may  be 
raised  to  the  power  required  in  a  more  simple  manner  than  by 
the  preceding  rule. 

Thus  let  it  be  required  to  raise  ^2  a  to  the  second  power. 
The  proposed  from  what  has  been  said,  art.  139,  may  be  put 

under  the  form  |     / sf2a;  but  to  raise  this  expression  to  the 

second  power,  it  is  sufficient  to  suppress  the  first  radical  sign; 
whence  (^/2  of  =  \/2  a. 

Again,  let  it  be  required  to  raise  \/5b  io  the  third  power. 
The  proposed  maybe  put  under  the  form  |    /  wdb  ;  whence 

Whence  if  the  index  of  the  radical  is  divisible  by  the  exponent 


FORMATION    OF    POWERS    AND   EXTRACTION    OF    ROOTS.         1^5 

of  the  poiuer,  to  ivkich  the  proposed  quantity  is  to  be  raised,  the 
operation  is  performed  by  dividing  the  index  of  the  radical  by  the 
exponent  of  the  power. 

144.  With  respect  to  the  extraction  of  roots,  it  is  evident,  from 
the  preceding  rules,  that  to  extract  the  root  of  a  radical,  we  may 
extract  the  root  of  the  qicaniity  placed  under  the  radical  sign, 
the  result  being  left  under  the  same  radical  sign,  or  we  may 
multiply  the  index  of  the  radical  by  the  index  of  the  root  to  be 
extracted. 

3 


Thus, 


^y^y^^=4/3^.  ^y^yrc^f^sF. 


EXAMPLES. 

1.  To  raise  ^a*b^  to  the  fourth  power. 

2.  To  raise  i>/ dh^c  to  the  sixth  power. 

3.  To  find  the  fourth  root  of  sfZ2a'b\ 

4.  To  find  the  fifth  root  of  >v/243^^ 

REDUCTION    OF    RADICAL    EXPRESSIONS    TO    THE    SAME 
INDEX. 

145.  It  follows  from  the  principles  established  above,  that 
if  we  multiply  at  the  same  time  the  index  of  the  radical  and  the 
exponents  of  the  quantity  placed  under  the  radical  sign  by  the 
same  number,  the  value  of  the  radical  remains  the  same. 

Thus  if  we  multiply  the  index  of  the  radical  s/c^b  by  3, 
we  have  wc^b,  the  third  root  of  the  proposed;  if  then  we 
multiply  the  exponent  of  the  quantity  placed  under  the  radi- 
cal sign  by  3,  we  have  />/a^b^  the  third  power  of  i^c^b;  the 
second  operation,  therefore,  restores  the  expression  to  its  original 
value. 

146.  By  means  of  this  last  principle,  we  may  reduce  two  or 


196  ELEMENTS    OF    ALGEBRA» 

more  radicals  of  different  indices  to  the  same  index.  Thus  let 
there  be  the  two  radicals  s/^a^  s/h^c.  Multiplying  the  index 
and  also  the  exponents  of  the  quantities  placed  under  the  radical 
sign  in  the  first  by  4,  and  in  the  second  by  3,  we  have  for  the 
first  v'SV  or  f^\^a\  and  for  the  second  ^¥J.  The  pro- 
posed are,  therefore,  reduced  to  equivalent  expressions  having  a 
common  index  12. 

In  like  manner,  the  three  quantities 

f^TW,  ^'dfW,  V7W, 
become  respectively 

105 105 105 

^/a'^V\   s/a'^y^  s/c^iS^ 
having  a  common  index  105. 

From  what  has  been  done  we  have  the  following  rule  for  re- 
ducing radical  expressions  to  the  same  index,  viz.  Multiply  at 
the  same  time  the  index  belonging  to  each  radical  sign,  and  the 
exponents  of  the  quantities  placed  under  this  sign,  by  the  product 
of  the  indices  belonging  to  all  the  other  radical  signs. 

If  the  indices  of  the  radicals  have  common  factors,  the  calcula- 
tions are  rendered  more  simple,  by  taking  for  the  common  index 
the  least  number  exactly  divisible  by  each  of  the  indices. 

A  quantity,  which  has  no  radical  sign,  may  on  the  same  prin- 
ciples be  placed  under  a  radical  sign ;  for  this  purpose,  we  raise 
the  quantity  proposed  to  the  power  denoted  by  the  index  of  the 
radical  sign,  under  which  it  is  to  be  placed. 
-  Thus  if  it  be  required  to  put  the  quantity  o^  under  the  sign 

^,  we  have  for  the  result  Va^". 

147.  Radical  expressions  having  different  indices  must  be  re- 
duced to  the  same  index  before  applying  to  them  the  rules  for 
multiplication  and  division  laid  down  above.  The  following 
examples  will  serve  as  an  additional  exercise  in  the  multiplica- 
tion and  division  of  radical  quantities. 


THEORY   OF    EXPONENTS.  197 

1.  Multiply  ^2  by  >,J/3.  Ans.  ^2592. 

2.  Multiply  V«  by  ^b.  Ans.  \/'^¥'. 

3.  Multiply  ^ahj  ^b.  Ans,  \^'^. 
'  4.  Multiply  3 a a/8^  by  2 bs/TH.  Ans,  12 a'^ i \/2Z 

5.  Multiply  V  2,  yv^  3,  and  ^  5  together.     Ans.  a^648000. 

6.  Multiply  ^f,  ;^J,  and  ^6  together,  Ans.  ^/y. 

7.  Multiply  2V6  —  3V5  by  4  VS  —  '^IO^ 

Ans.  39V2— 16V15. 

a  Multiply  4  V^  +  5 V^  by  Vi  +  2a/^-  _« 

Ans.  '^j^  +  V^  V42. 

9.  Divide  ></6  by  ^2.  Ans.  J/v'55296: 

10.  Divide  \/l35  by  V^.  Ans.  a/ts: 

11.  Divide  aX/b  by  ^^.  Ans.  a^^b, 

12.  Divide  ^«  by  s/b  ■\-  s/c.  Ans. . 

0  —  c 

13.  Divide  V  3  by  1  +  V  2.  Ans.  V  3  (V  2  —  1). 

14.  Divide  1  Vi  by  V  2  +  3  Vi-  Ans.  ^n- 


15.  Divide  1  by  Va  -j-  V**  Ans 

16.  Divide 


Y        a^  —  b  ' 


a/s/^^  +  s/^  by  ^/s/a  —  s/b, 

A„s.  »/^±ll2V5 

r  a  —  b 


SECTION  XIX.— Theory  of  Exponents. 

148.  We  have  seen,  art.  28,   that  with  respect  to  the  same 
letter,  division   is  performed  by  subtracting   the   exponent   of 


198  ELEMENTS    OF    ALGEBRA. 

the  divisor  from  that  of  the  dividend.  The  application  of  this 
rule  to  the  case,  in  which  the  exponent  of  the  divisor  is  equal  to 
that  of  the  dividend,  gives  rise  to  the  exponent  0.  An  expres- 
sion a",  in  v/hich  this  exponent  is  found,  is  to  he  regarded, 
art.  31,  as  a  symbol  equivalent  to  unity, 

149.  The  application  of  the  same  rule  to  the  case,  in  which 
the  exponent  of  the  divisor  exceeds  that  of  the  dividend,  gives 
rise  to  negative  exponents.  Thus  let  it  be  required  to  divide 
a^  by  a^.  Subtracting  the  exponent  of  the  latter  from  that  of  the 
former,  we  have  a~^  for  the   result.     But  a^  divided  by  c^  is 

expressed  by  the  fraction  -g ;  reducing  this  fraction  to  its  lowest 
terms,  we  have  —^.  The  expression  a~'^  must  therefore  be  re- 
garded as  equivalent  to  —^. 

In  like  manner  ^^_^_,^  gives  by  subtracting  the  exponent  of 
the  divisor  from  that  of  the  dividend  a~";  but  the  fraction 
fifives  when  reduced  to  its  lowest  terms  — ;   whence  a"" 

is  equivalent  to  — . 

The  expression  «""  is,  therefore,  the  symbol  of  a  division, 
which  cannot  be  performed.  Its  true  value  is  the  quotient  of 
unity  divided  by  a  raised  to  a  -power  denoted  by  the  negative 
exponent  n. 

150.  To  find  the  roots  of  monomials,  we  divide,  art.  122,  the 
exponents  of  the  proposed  by  the  index  of  the  root  required. 
The  application  of  this  rule  to  the  case,  in  which  the  exponents 
of  the  proposed  are  not  divisible  by  the  index  of  the  root,  gives 
rise  to  fractional  exponents.  Thus  let  the  third  root  of  a  be 
required.  Indicating  upon  the  exponent  of  a  the  operation  re- 
quired in  order  to  obtain  the  third  root,  we  have  for  the  re- 
sult a^.     But  we  have  agreed  to  indicate  the  third  root  by  >^; 


THEORY    OF    EXPONENTS.  199 

the  expressions  /^  a,  a^  are,  therefore,  to  be  regarded  as  equiva- 
lent.    In  like  manner,  we  have 

m 

The  expression  a"  is,  therefore,  to  be  regarded,  as  a  symbol 
equivalent  to  the  nth  root  of  the  mth  power  of  a. 

151.  The  two  preceding  cases  sometimes  meet  in  the  same 
expression.  This  gives  rise  to  negative  fractional  exponents. 
Thus  let  it  be  required  to  extract  the  seventh  root  of  a?  di- 

vided  by  a^ ;  we  have  -g  =  a~^  the  seventh  root  of  which  is 
a  ~  T.     In  like  manner  the  wth  root  of 


The  expression  a  "  is,  therefore,  the  symbol  of  a  division 
which  cannot  be  performed,  combined  with  the  extraction  of  a 
root.  Its  true  value  is  the  nth  root  of  the  quotient  of  unity  di- 
vided by  a  raised  to  the  mth  power. 

m  m 

The  expressions  a",  «"'",  a",  a  ",  derived  in  the  manner 
above  explained  from  rules  previously  established,  have  be- 
come   by    agreement    notations    equivalent    respectively   to    1, 

1      ""  y  I 

— ,  /^(f"^    t  /    — ;   we  may,  therefore,  at  pleasure  substitute 

the  former  of  these  expressions  for  the  latter,  and  the  converse. 
152.  We  proceed  to  show,  that  the  rules  already  established 
for  performing  the  operations  of  arithmetic  upon  quantities 
affected  with  entire  and  positive  exponents  are  sufficient  for 
these  operations,  whatever  the  exponents  may  be,  with  which 
the  quantities  are  affected. 


200  ELEMENTS    OF    ALGEBRA 


MULTIPLICATION. 


Let  it  be  required  to  multiply  a^  hy  a  .     To  perform  the 
Operation  required,  it  is  sufficient  to  add  the  exponents. 

Indeed  J  =  ^^,  J  =  a/^,  whence 
But  adding  the  exponents,  we  have 

,fxJ  =  .i  +  i  =  e« 

the  same  result  as  before. 

Again,  let  it  be  required  to  multiply  a~^  by  a*;  we  have 

whence   a~^  X  «^  =  l/^^3  X  ^7^^=  tV'^  X  )^a}'  .  . 

But  adding  the  exponents  of  the  proposed,  we  have 
the  same  result  as  by  the  former  operation. 

m  p 

Let  it  be  required  next  to  multiply  a     "  by  <z'  ; 
we    have      a    '^  =  \X   '^t>       a?  =  t^oF  ; 


whence 


nq       /■ 


We  arrive  at  the  same  result  by  adding  the  exponents  of  the 
proposed. 

_  2!      I      Z.  np  —  mq 

Indeed         a   ""        '  =a    '"'    . 


THEORY   OF   EXPONENTS.  201 

To  multiply  two  monomials  therefore,  it  is  sufficient,  whatever 
the  exponents,  to  add  the  exponents  of  the  letters^  which  are  the 
same  in  each. 

EXAMPLES. 

1.  Multiply  a*  c*,  a ""  ^  3,  and  c^ i "" ^  together. 

Ans.  a"^^^, 

2!  Multiply  -y^  by  ^.  Ans.  a^^bK^^. 

3.  Multiply  a^  +  b^  by  a^  —  b^.  Ans.  a^  —  b^, 

4.  Multiply  3  +  52  by  2  —  5*.  Ans.  1  —  sK 


DIVISION. 

153.  Whatever  the  exponents  may  be,  in  order  to  divide  one 
monomial  by  another,  we  subtract  for  each  letter  the  expon£nt 
of  the  divisor  from  that  of  the  dividend. 

Indeed,  since  the  exponent  of  each  letter  in  the  quotient  should 
be  such,  that  when  added  to  the  exponent  of  the  same  letter 
in  the  divisor,  the  sum  will  be  equal  to  the  exponent  of  the  divi- 
dend, it  follows,  that  the  exponent  of  the  quotient  should  be  equal 
to  the  difference  between  that  of  the  divisor  and  the  dividend. 

EXAMPLES. 


1.  Divide  a'  by  a"*. 

Ans.  a^. 

2.  Divide  a*  by  a^. 

Ans.  a"^\ 

3.  Divide  a^b^  by  a''^b^. 

Ans.  a"^^*"*. 

4.  Divide  a^  —  b^  by  a*  —  i*.       Ans.  a^  +  ah^  +  b^. 

5.  Divide  5a^  — 41a^i  +  42fl^i«  by  5a^  —  eah. 

Axis,  fli  — y-""?'- 


ELEMENTS    OF    ALGEBRA. 
FORMATION    OF    POWERS   AND   EXTRACTION   OF   ROOTS. 

154.  From  the  rule  for  multiplication,  it  follows,  that  to  raise 
a  monomial  to  any  power,  it  is  necessary  whatever  the  exponents 
of  the  letters,  to  multiply  the  exponent  of  each  letter  by  the 
exponent  of  the  power  required. 

_g 
Thus  a       raised  to  the  third  power 

_,-i+(-i)^(-i)_,-«_„-'. 

Conversely,  to  extract  the  root  of  a  monomial,  we  dividb  the 
exponent  of  each  letter  by  the  index  of  the  root. 

Thus  ^a-2  =  a"^ 

The  utility  of  exponents,  of  the  kind  which  we  are  here  con- 
sidering, consists  principally  in  this,  that  the  calculation  of  quan- 
tities affected  with  these  exponents  is  performed  by  the  rules 
already  established  for  quantities  affected  with  entire  and  positive 
exponents.  The  calculation  is  moreover  reduced  to  operations 
upon  fractions,  with  which  we  are  already  familiar. 

155.  By  means  of  negative  exponents  we  may  give  an  entire 
form  to  fractional  expressions.     Thus,  let  there  be  the  fraction 

-5,    this    is    the    same    as    a;  X  -5  J    t)ut    —  =  2/  ~  ^ ;    whence 
r  f  f 

^  -a 

156.  Fractional  and  negative  exponents  enable  us  to  arrange 
polynomials,  which  contain  radical  terms.  Thus  let  it  be  re- 
quired to  arrange  the  polynomial 

according  to  the  descending  powers  of  the  letter  a. 

To  perform  the  operation  required,  1°.  we  give  to  the  radical 
quantities  fractional  exponents ;  2^.  we  reduce  to  an  entire 
form,  terms  which  have  denominators;  3°.  we  reduce  all  the 
exponents  of  the  letter,  according  to  which  the  arrangement  is  to 
be  made,  to  their  least  common  denominator.     The  proposed 


PROPORTION   BY   DIFFERENCE.  269 

may  then  be  arranged  according  to  the  powers  of  the  letter 
required. 

In  the  preceding  example  we  have  for  the  result 


SECTION  XX.— Proportions. 


157.  When  two  quantities  are  compared  with  respect  to  their 
magnitude,  the  result  of  the  comparison  is  called  their  ratio.  In 
general,  there  are  two  different  ways,  in  which  the  magnitude 
of  two  quantities  may  be  compared;  P.  we  may  wish  to  de- 
termine how  much  the  greater  exceeds  the  less;  the  result  is 
then  obtained  by  subtraction,  and  is  called  the  ratio  of  the 
quantities  by  difference;  2°.  we  may  wish  to  determine  how 
often  one  of  the  quantities  is  contained  in  the  other ;  the  result  is 
then  found  by  division  and  is  called  the  ratio  of  the  quantities 
by  quotient. 

Thus,  the  ratio  by  difference  of  the  quantities  a  and  b  is 

a  —  b,  and  the  ratio  by  quotient  is  -;  a  and  b  are  the  terTns 

of  the  ratio. 

The  same  quantity  may  be  added  to,  or  subtracted  from,  both 
terms  of  a  ratio  by  difference  loithout  changing  the  ratio,  for 
a  —  b={a-\-c)  —  {b-\-c)  =  {a  —  c)  —  [b  —  c). 
TJie  two  terms  of  a  ratio  by  quotient  may  be  multiplied  of 
divided  by  the  same  quantity  without  changing  the  ratio,  for 
a       am 
b       bm' 
Ratios  by  difference  are  sometimes  called  arithmetical  ratios 
and  those  by  quotient  geometrical  ratios. 

158.  An  expression  for  two  equal  ratios  is  called  a  proportion* 


204  ELEMENTS    OF    ALGEBRA. 

If  the  ratios  are  by  difference,  the  proportion  is  called  a  propor 
tion  by  difference.     Thus  the  equality 

b  —  a  =  d  —  c, 
is  a  proportion  by  difference,  and  is  usually  written  thus, 
a  .b:c  .  d. 
If  the  ratios  are  by  quotient,  the  proportion  is  called  a  propor- 

CL  C 

tion  by  quotient.     Thus,  the  equality  j  =  -  is  a  proportion  by 

0  CL 

quotient,  and  is  usually  written 

a:b::c:d. 

The  proportions  above  are  read  thus,  a  is  to  i  as  c  to  d.  The 
first  and  last  terms  are  called  the  extremes  of  the  proportion; 
the  second  and  third  are  called  the  means ;  a  is  called  the  ante- 
cedent, b  the  consequent  of  the  first  ratio ;  c  the  antecedent,  d  the 
consequent  of  the  second  ratio. 

Proportion  by  difference  is  sometimes  called  arithmetical  pro- 
portion, that  by  quotient  geometrical  proportion.  Proportion  by 
difference  is  now,  however,  more  commonly  called  equidiffer- 
ence,  while  the  term  proportion  is  limited  to  proportions  by 
quotient. 

EQUIDIFFERENCES. 

159.   Let  there  be  the  equidifference  a.b:c.d\  this  is  the 
same  with  the  equation  b  —  a  =  d  —  c,  from  which  we  deduce 
a -\- d  =z  b  -\- c. 

Thus  in  an  equidifference  the  sum  of  the  extremes  is  equal  to 
the  sum  of  the  means.  This  is  the  leading  property  of  equi- 
differences. 

Reciprocally,  let  there  be  four  quantities  a,  b,  c,  d,  such  that 
a-\-  d  =  b  -^-c.     From  this  equation  we  obtain 
b  —  a  =  d  —  c,  or  a  .  b  :  c  .  d. 

Thus,  if  there  be  four  quantities  such,  that  any  two  of  them 
give  the  same  sum  with  the  other  two,  the  first  are  the  extremes ^ 
the  second  the  mearis,  or  the  converse,  of  an  equidifference. 


PROPORTION  BY  QUOTIENT.  205 

Any  three  terms  of  an  equidifference  are  sufficient  to  deter- 
mine the  fourth ;  thus,  from  the  equidifference,  a  .b  :c  ,  dj  we 
deduce  a  =  b  —  d  -{-  c,  b==.a-\-  d  —  c. 

In   the   equidifference   a  .b:  c  .  d,   let  c  =  3 ;    we   have 
a  .b  :b  .  d. 

This  is  called  a  continued  equidifference,  and  b  is  called  an 
arithmetical  mean  between  a  and  d. 

From  the  equidifference  a  .b  :b  .  d,  we  deduce 
b  =  \{a  +  d); 
the  arithmetical  mean  betioeen  tivo  quantities  is,  therefore,  equal 
to  half  their  sum. 

160.  In  order  that  an  equidifference  may  exist,  it  is  sufficient, 
that  the  sum  of  the  extremes  should  be  equal  to  the  sum  of  the 
means ;  we  may,  therefore,  make  any  transposition  of  the  terms, 
of  an  equidifference,  which  will  not  alter  the  equality  between 
whe  sum  of  the  extremes  and  that  of  the  means.  The  equation 
a  —  b  =  c  —  d  furnishes  the  eight  following  equidifferences, 

a  .  b  :  c  .  d,   a  .  c:  b  ,  d,   d  ,  b  :  c  .  a,   d  .  c:  b  .  a, 
h  .  a:  d  .  c,   b  .  d:  a  .  c,   c  .  d:  a  .b,   c  .  a:  d  .b. 

PROPORTION    BY    QUOTIENT. 

161.  Let  us  take  the  proportion  a:b:  :  c:d;  this  returns 
o  -=i-,  an  equation,  which  gives 

adi=bc. 

In  a  proportion  by  quotient,  therefore,  the  product  of  the  eX' 
tremes  is  equal  to  the  product  of  the  means.  This  is  the  funda- 
mental property  of  proportions. 

Reciprocally,  let  there  be  four  quantities  a,  b,  c,  d,  such,  that 

ad  =  bc;  this  leads  to  the  equation  -  =  -,  or 

a       c 

a:  bit:  d. 

Whence  if  four  quantities  be  such,  that  any  two  of  them  give  the 

tame  product  as  the  remaining  two,  the  first  will  form  the  ex* 

iremes  and  the  second  the  means^  or  the  converse,  of  a  proportion. 


206  ELEMENTS    OF    ALGEBRA. 

Three  terms  of  a  proportion  are  sufficient  to  determine  the 

fourth ;  thus  from  the  proportion  a:h  '.:  c:  d^  we  deduce 

he    ^       ad    ^ 
a=^-rr,  b  =  — ,  &c. 
a  c 

The  proportion  a:  b  : :  b:  d,  in  which  the  two  mean  terms  are 
the  same,  is  called  a  continued  proportion,  and  b  is  called  a  rnean 
proportional  between  a  and  d. 

From  the  continued  proportion  a:  b  :  :  b  :  d,vfe  deduce  P=:ad, 

whence  br=W  ad.     Thus  to  find  a  mean  proportional  betioeen 
two  quantities,  we  take  the  square  root  of  their  product. 

162.  In  order  that  a  proportion  may  exist,  it  is  sufficient,  that 
the  product  of  the  extremes  should  be  equal  to  that  of  the  means. 
We  may,  therefore,  make  any  transposition  in  the  terms  of  a 
proportion,  which  will  leave  the  product  of  the  extremes  equal  to 

that  of  the  means.     Thus  the  equation  -  =  -  gives  the  eight 

following  proportions 

a  :  b  :  :  c  :  d,   a  :  c  :  :  b  :  d,   b  :  d  :  :  a  :  c,   d  :  c  :  :  b  :  a, 
b  :  a  :  :  d  :  c,   c  :  a:  :  d  :  b,   d  :  b  :  :  c  :  a,   c  :  d  :  :  a  :  b. 

163.  The  same  quantity  m,  it  is  evident,  may  be  added  to  or 

subtracted  from  the  equation  -  =  -,  so  that  we  have 

a       c 

b^  d^ 

.    a  c 

,                                  b-:^ma       d-±:mc 
whence  = . 

a  c 

but  this  last  may  assume  the  form 

c       d-^mc 

a       b  z^ma' 

from  which  we  have  the  proportion 

b  :^ma\  d-:^mc\  :  a:c; 

c       d 
*ut  since  -  =  T>  we  have  also 
a      0 

d       d-±z  "fnc 
b'     h-^ma 


PROPORTION  BY  QUOTIENT.  207 

from  which  we  have  the  proportion 

h  -i^ma  :  d  ±  ^c  :  :  b  :  d. 

These  two  proportions  may  be  enunciated  thus ;      The  first 

consequent  plus  or  minus  its  antecedent  taken  a  given  number  of 

times,  is  to  the  second  consequent  plus  or  minus  its  antecedent 

taken  the  same  number  of  times,  as  the  first  term  is  to  the  third, 

or  as  the  second  is  to  the  fourth. 

^aA     m.  .       d±:mc        c 

lo4.    1  he  expression  ~ =  -  returns  to 

0  ±  'ma       a 


d-\-mc        c    d 


c 


b-\-ma       a    b  —  ma       a 

.  d~\-mc       d  —  mc 

whence  ,— y =  -, , 

b-\-ma        b  —  ma 

or  b -\- ma:  d-\-mc::b  —  ma:d  —  mc, 

or,  changing  the  relative  places  of  the  means, 

b  -\-ma:b  —  ma'.  '.  d-\-mc:  d  —  mc] 

whence  making  m=^\,  we  have 

b  -\-  a:b  —  a::  d-\-  c  d  —  c, 

a  proportion  which  may  be  enunciated  thus. 

The  sum  of  the  first  two  terms  is  to  their  difference,  as  the  sum 

of  the  last  two  is  to  their  difference. 

165.   The  proportion  a  :  b  :  :  c:  d  may  be  written  thus, 

a  :  c  :  :  b  :  d, 

we  have  then 

c  d 

a  b 

from  which  we  obtain 

^  c-^zma-.d^mb::  a-.b,  OT  ::  c:  d; 

whence,  the  second  antecedent  plus  or  minus  the  first  taken  a 

given  number  of  times,  is  to  the  second  consequent  plus  or  minus 

the  first  taken  the  same  number  of  tim£s,  as  any  one  of  the  ante' 

cedents  whatever  is  to  its  consequent. 

If  in  the  above  proportion  we  make  m=  1,  we  have 

c ':^ a  :  d :^b  :  :  a  :  b,  ox  '. :  c  :  d ', 

whence  c-\-a:c  —  a::d-\-b'.d  —  h. 


SOS  ELEMENTS    OF    ALGEBRA. 

Therefore,  the  sum  or  difference  of  the  antecedents  is  to  the  sum 
or  difference  of  the  consequents,  as  one  antecedent  is  to  its  consc' 
quent ;  and  the  sum  of  the  antecedents  is  to  their  difference^  as  the 
sum  of  the  consequents  is  to  their  difference. 

166.  Let  there  be  the  series  of  equal  ratios 

a:  b  :  :  c:  d:  :  e  :/:  :  g  :  h  ,  .  .  , 

^__d__f_h 
a       c        e       g' 

Making  —  =  q;  we  have 

b  d  f  h 

-  =  q,-  =  q,-  =  q,-=zq; 

a  c  e  g 

whence  bz=aq,  d  =  cq,  fz=eq,  h  =  gq; 

adding  these  equations  member  to  member,  we  have 

b  +  d+f+h  =  {a  +  c  +  e  +  g)q; 

,         •  b+d+f+h  b 

whence  — ' r^ —  =  <7  =  -» 

a-f-c  +  e  +  ^  a 

or  a-{-c-\-e-^  g:  b-{-d-{-f-^h::a:b; 

whence  in  a  series  of  equal  ratios,  the  sum  of  any  number  what' 

ever  of  antecedents,  is  to  the  sum  of  the  like  number  of  consC' 

quents,  as  one  antecedent  is  to  its  consequent. 

167.  Let  there  be  the  two  equations  -  =  -,--=-;  muhiply- 

a       c  e       g 

ing  these  equations,  member  to  member,  we  have 
bj__dh 
ae       cg^ 
or  ae  :  bf: :  eg  :  dh. 

We  obtain  the  same  result  by  multiplying,  term  by  term,  the 
proportions  a:b  :  :  c:  d,  e:f::g:h;  this  is  called  multiplying 
the  proportions  in  order  ;  it  follows  then,  that  if  two  proportions 
be  multiplied  in  order,  the  results  will  be  proportional. 

It  will  be  seen  also,  that  if  two  proportions  be  divided,  term  by 
term,  or  in  order^  the  quotients  will  be  proportional. 


PROPORTION  BY  QUOTIENT.  209 

If  in  the  equation  -  =  -  we  raise  both  members  to  the  mth 
a       c 

power,  we  have 

3"* dr 

which  gives  aP" :  b^  :  :  c^  :  dl^. 

It  follows,  therefore,  that  the  second^  thirds  and  in  general  the 
similar  powers  of  four  proportional  quantities  are  also  propor- 
tional. 

In  like  manner,  it  may  be  shown  that  the  roots  of  the  same 
degree  of  four  proportional  quantities  are  also  proportional. 

QUESTIONS    IN   WHICH    PROPORTIONS   ARE    CONCERNED. 

1.  The  sum  of  the  squares  of  two  numbers  is  to  the  difference 
of  their  squares  as  17  to  8,  and  their  product  is  15.  What  are 
the  numbers  ? 

Putting  X  and  y  for  the  numbers,  we  have  by  the  first  con- 


dition 

^  +  f 

7?- 

-7/::17:8; 

whence  art.  164, 

22? 

22/^::  25:  9, 

or 

a? 

2/^::25:9; 

whence  art.  167, 

X 

y  '.:    5:3, 

wherefore 

3z=5v. 

By  the  second  condition  we  have  xy=\5',  comparing  this 
w:th  the  equation  just  found,  we  readily  obtain  a:  =  5,  ?/  =  3. 

2.  The  product  of  two  numbers  is  24,  and  the  difference  of 
their  third  powers  is  to  the  third  power  of  their  difference  as 
19  to  1.     Required  the  numbers. 

Let  X  and  y  be  the  numbers,  we  have  by  the  question 

a:y  =  24 
7?  —  'if:{x--yf:'.\^'.\. 

Transposing  terms  and  developing  {x  —  y)^  in  this  last,  we 

have  7?-'f:\^:'.7?—^7?y-{-^xif  —  f'.\\ 

whence  art.  165,         ^y?y  —  2xy^ :  18  : :  (a:  —  yf  :  1, 

or  dividing  by  a:  —  y  and  substituting  for  xy  its  value  from  the 

first  equation  72  :  18  :  :  (a:  —  y)' :  1 ; 

whence  z  —  v  =  2. 

U  ^ 


210  ELEMENTS    OF    ALGEBRA. 

Comparing  this  last  with  the  first  equation  we  obtain 
a;  =  6,  ?/  =  4. 

3.  The  sum  of  two  numbers  is  to  their  difference  as  3  to  1, 
and  the  difference  of  their  third  powers  is  56;  what  are  the 
numbers  ?  Ans.  4  and  2. 

4.  There  are  two  numbers,  whose  product  is  135,  and  the 
difference  of  their  squares  is  to  the  square  of  their  difference 
as  4  to  1.     What  are  the  numbers  ?  Ans.  15  and  9. 

5.  A  merchant  mixes  wheat,  which  cost  him  10  shillings  a 
bushel,  with  barley  which  cost  him  4  shillings  a  bushel,  in  such 
proportion  as  to  gain  43J  per  cent,  by  selling  the  mixture  at 
11  shillings  a  bushel.     Required  the  proportion. 

Ans.  14  bushels  of  wheat  to  9  of  barley. 

6.  The  product  of  two  numbers  is  63,  and  the  square  of  their 
sum  is  to  the  square  of  their  difference  as  64  to  1.  What  are 
the  numbers  ?  Ans.  9  and  7. 


SECTION  XXL— Progressions. 

168.  A  series  of  quantities  increasing  or  decreasing  by  a  con- 
stant difference,  is  called  an  arithmetical  progression  or  prO' 
gression  by  difference.  The  constant  difference  is  called  the 
ratio  of  the  progression. 

Thus  let  there  be  the  two  following  series 
1,    4,    7,  10,  13,  16 
60,56,52,48,44,40. 
the  first  is  called  an  increasing  progression,  the  ratio  of  which 
is  3 ;  the  second  is  called  a  decreasing  progression,  the  ratio  of 
which  is  4. 

To  indicate   that   the   quantities  c,  h,  c,  d  .  .  .  form  a  pro- 
gression by  difference,  we  write  them  thus 
-^  a  .b  ,c  .d  .  .  ,  , 
A  progression  by  difference,  it  will  be  readily  perceived,  is 


PEOGRESSION    BY   DIFFERENCE.  5^| 

simply  a  series  of  continued  equidifferences.  Each  term  there- 
fore is  at  once  antecedent  and  consequent,  with  the  exception  of 
the  first  term,  which  is  only  an  antecedent,  and  of  the  last,  which 
is  only  a  consequent.  The  progression  -f-  a  .  i  .  c  .  d  is  enun- 
ciated thus,  a  is  to  i  as  3  to  c,  as  c  to  d^  &cc. 

169.  Let  us  take  the  increasing  progression 

•7-a.b.c.k.h.., 
and  let  d  represent  the  ratio. 

From  the  nature  of  the  progression,  we  have,  it  is  evident, 
b=a  -|-  d 
c==a-\-2d 
k  =  a-!rSd, 
from  which  it  is  readily  inferred,  that  a  term  of  any  rank  lohat- 
ever  is  equal  to  the  first  term  plus  as  many  times  the  ratio,  as 
there  are  units  in  the  number  of  the  preceding  terms. 

Let  L  represent  a  term  of  any  rank  whatever,  and  let  n  denote 
the  number,  which  marks  the  place  of  this  term ;  we  have  from 
what  has  been  said 

L  =  a-\-  {n —  l)d. 
This  expression  for  L  is  called  the  general  term  of  the  series. 
If  the  series  were  decreasing,  we  should  have,  as  it  is  easy  to 
see,  for  the  general  term 

L  =  a  —  {n  —  l)d. 
By  means  of  the  above  formulas  we  may  find  any  term  of  a 
progression  by  difTerence,  when  the  first  term,  the  number  of  the 
term  required,  and  the  ratio  are  given. 

Thus,  let  it  be  required  to  find  the  50th  term  of  the  pro- 
gression -T-  1 .  4 .  7 ,  we  have  by  the  first  formula 

I<=1  + (50  — 1)3=148. 
Again  let  it  be  required  to  find  the  40th  term  of  the  pro- 
gression -t-  5 . 3 .  1  .  .  .,  we  have  by  the  second  formula 
L==5— (40  — 1)2  =  — 73. 

170.  The  first  and  last  terms  of  a  progression  are  called  the 
extremes;  if  the  number  of  terms  be  odd,  the  middle  term  is 
called  the  mean;   if  the  number   of  terms   be   even,  the  two 


212  ELEMENTS    OF   ALGEBRA. 

terms  having  the  same  number  of  terms  on  each  opposite  side 
are  called  the  means. 

Let  us  take  the  general  progression  -^  a  .h  .c  . , ,  ,h  .h  .1, 
from  the  nature  of  the  progression,  we  have 

h  —  a  =  l  —  A 
whence  b  -\-k-=a-\-l 

so  also  c  —  b  =  k  —  n 

whence  c-|-A  =  i-|-Ar  =  «4-Z, 


from  which  we  infer  that  in  a  progression  by  difference,  the  sum 
of  any  two  terms  taken  at  equal  distances  from  the  extremes  is 
equal  to  the  sum  of  the  extremes. 

Let  S  represent  the  sum  of  all  the  terms  in  the  progression 
■^  a  .b  .  c  . .  .  ,k  .  k  .1,.  Writing  this  progression  in  an  inverse 
order  below  itself,  we  have 

S  =  a-{-b  +  c  ....  k  +  k-^l 
S=l-\-k  +  h  ....  c-i-b-i-a; 
adding   these   equations   member   to   member   and  uniting   the 
corresponding  terms,  we  have 

2S={a  +  l)  +  {b  +  k)....  +  {h  +  c)  +  {k  +  b)  +  {l  +  a) 
but  the  parts  b-{-kf  k-j-c  .  .  .  are  equal  each  to  a  +  Z ;  the 
number  of  these  parts  moreover  is  the  same,  it  is  evident,  with 
the  number  of  terms  in  the  progression;  designating  then  this 
number  by  «,  we  have 

2S  =  n{a-{-l); 

whence  S^^^pl. 

By  means  of  this  formula  we  find  the  sum  of  all  the  terms, 
when  the  first  term,  the  last  term  and  the  number  of  the  terms 
are  given. 

EXAMPLES. 

1.  What  is  the  sum  of  the  natural  series  of  numbers  1,  2, 
3,4,  &c.  up  to  1000? 

2.  The  last  term  in  a  progression  by  difference  is  60,  the  first 
term  12,  and  the  number  of  terms  5.  Wliat  is  the  sum  of  all 
the  terms  ? 


PROGRESSION   BY    DIFFERENCE. 


213 


3.  What  is  the  sum  of  the  uneven  numbers  1,  3,  5,  7,  &c. 
up  to  93  ? 

171.  The  equations  L  =  a-{- {n—l)d,  g^^(g  +  [)  f^^. 

nish  us  with  the  means  of  resolving  the  following  general 
problem,  viz.  Any  three  of  the  five  quantities,  a,  d,  n,  1  and  s, 
which  enter  into  a  progression  by  difference,  being  given,  to  dc' 
termin£  the  remaining  two. 

This  general  problem  resolves  itself  into  as  many  particular 
problems  as  there  are  combinations  of  5  letters  taken  2  and  2,  or 
3  and  3  at  a  time.  The  number  will  therefore  be  10.  See  the 
enunciations  below. 

Let  there  be  given   1°.  a,  d,  n  to  find  I  and  *  ^  ^      ■'      I 

2°.  a,  d,  I   ...  n  and  s  -^ 

S°.  a,  d,  s   .    .    .  n  and  I 

4°.  a,n,  I  .    .   .  d  and  s 

5°.  a,  n,  s   .    .    .  d  and  I 

6°.  a,  I,  s   .    .    .   d  and  n 

7°.  dfU,  I   .    .    .  a  and  i 

B°.  d,  n,  s  .    .    .a  and  I 

9°.  d,  I,  s   .    .    .  «and» 
10°.  n,l,s...  «andrf^i>^^  .   - 

172.  Let  it  now  be  proposed  to  find  the  number  of  terms  in 
the  progression  by  difference,  the  sum  of  which  is  145,  the  first 
term  1,  and  the  ratio  3.  ^    .  — 

This  example  is  a  particular  case  of  the  third  general  prob- 
lem ;  to  prepare  a  formula  for  its  solution,  we  eliminate  L  from 

the  equations 

71  {a  -\- 1) 
2       ' 


L  =  a-\'{n—\)d,         S  = 

by  which  we  obtain  for  the  equation  for  n 
2       {d  —  2a) 


d 


2£ 

d' 


resolving  this  equation  we  have 

__d  —  2a±^/{d  —  2af-{-Sds 
"*""  2d 


214  ELEMENTS    OF    ALGEBRA. 

from  which  we  obtain  10  for  the  number  of  terms  sought.  This 
being  done  we  readily  obtain  by  means  of  the  formula  for  L  the 
last  term  required. 

Let  the  learner  prepare  the  formulas  and  solve  the  following 
particular  cases. 

1.  To  find  the  first  and  last  terms  in  a  progression  by  differ- 
ence, the  sum  of  which  is  567,  the  number  of  terms  21  and  the 
ratio  2. 

2.  The  sum  of  a  progression  by  difference  is  1455,  the  first 
term  5,  and  the  number  of  terms  80.  What  is  the  last  term  and 
the  ratio  ? 

3.  The  first  term  in  a  progression  by  difference  is  5,  the  last 
term  185,  and  the  ratio  6,  to  find  the  number  of  terms,  and  the 
sum  of  all  the  terms. 

4.-  The  first  term  of  a  progression  by  difference  is  3,  the  last 
term  41,  and  the  sum  of  all  the  terms  440.  What  is  the  num 
ber  of  terms  and  the  ratio  ? 

173.  The  formula  L  =  a-\-  {n  —  1)^  gives  d  = =■ ;  this 

expression  for  d  enables  us  to  resolve  the  following  problem,  viz. 
to  insert  between  two  quantities,  b  and  c,  m  arithmetical  means, 
that  is  to  say,  quantities,  which  comprised  between  b  and  c,  will 
form  with  them  a  progression  by  difference. 

To  resolve  this  problem,  it  will  be  sufficient  to  ^determine  the 
ratio  of  the  progression  required.  For  this  we  have  given  the 
first  term  b,  the  last  term  c,  and  the  number  of  terms  m  -{-  2. 
Substituting  therefore  c,  b,  and  m  -|-  2  for  I,  a,  and  n  in  the 

above  expression  for  d,  viz.  d  = -,  the  ratio  required  will 

be  r— -= — -r-— .,  that  is,  to  find  the  ratio  sought,  we 

m-\-2  —  I       m-f-1 

divide  the  difference  of  the  two  numbers  b  and  c  by  the  number  of 

terms  to  be  inserted  plus  1. 

Let   it  be   required,  for   example,   to   insert   11  arithmetical 

means  between  17  and  77. 

77 17 

Here  d  =  — — —  =  5. 


PROGRESSION    BY    DIFFERENCE.  ^ISj 

The  progression  required  will  therefore  be 

-r  17  .  22  .  27  .  32 72  .  77. 

Let  it  be  required,  as  a  second  example,  to  insert  9  arithmet- 
ical means  between  each  antecedent  and  consequent  of  the  pro- 
gression -J-2.5.8.11.14.... 

It  will  readily  be  inferred  from  what  has  been  done,  that,  if 
between  the  terms  of  a  progression  by  difference^  taken  two  and 
two^  we  insert  the  same  number  of  arithmetical  means^  the  terms 
of  this  progression  together  with  the  arithmetical  means  iriserted 
win  form  a  progression  by  difference, 

QUESTIONS    INVOLVING    PROGRESSIONS    BY    DIFFERENCE. 

1.  A  number  consisting  of  3  digits,  which  are  in  arithmetical 
progression,  being  divided  by  the  sum  of  its  digits  gives  a  quo- 
tient 48;  and  if  198  be  subtracted  from  it  the  digits  will  be 
inverted.     Required  the  number. 

Let  X  =  the  second  digit  and  y  the  common  difference,  the 
three  digits  will  then  be  expressed  hy  x  ■-\-  y,  x,  x  —  y. 

Resolving  the  question  we  obtain  a:  =  3 ;  and  the  number  re- 
quired is  432. 

2.  Four  numbers  are  in  arithmetical  progression.  The  sum 
of  their  squares  is  equal  to  276,  and  the  sum  of  the  numbers 
themselves  is  equal  to  32.     What  are  the  numbers  ? 

Let  2?/=  the  common  difference  and  let  x-\-Sy,  x-\-y, 
X  —  y,x  —  3  y  be  the  numbers. 

Resolving  the  question,  we  obtain  for  the  numbers  sought, 
11,  9,  7  and  5. 

3.  A  traveller  sets  out  for  k  certain  place  and  travels  1  mile 
the  first  day,  2  the  second  and  so  on.  In  5  days  afterwards 
another  sets  out  and  travels  12  miles  a  day.  How  long  and  how 
far  must  he  travel  to  overtake  the  first  ? 

Let  X  =  the  number  of  days ;   then  a;  -|-  5  =  the  number  of 

a;  4-5 
days   the    first    travels,   and    {x-\-  6)  — ^ —  =  the   distance   he 

travels. 


216  ELEMENTS    OF    ALGEBRA. 

Resolving  the  question,  we  obtain  a;  =  3,  or  10. 

4.  There  are  three  numbers  in  arithmetical  progression, 
whose  sum  is  21 ;  and  the  sum  of  the  first  and  second  is  to 
the  sum  of  the  second  and  third  as  3  to  4.  Required  the 
numbers.  .  Ans.  5, 7,  9. 

5.  From  two  towns,  which  were  168  miles  distant,  two  per- 
sons, A  and  B,  set  out  to  meet  each  other;  A  went  3  miles  the 
first  day,  5  the  next,  and  so  on ;  B  went  4  miles  the  first  day, 
6  the  next,  and  so  on.     In  how  many  days  did  they  meet  ? 

Ans.  a 

6.  There  are  four  numbers  in  arithmetical  progression,  whose 
sum  is  28,  and  their  continued  product  is  585.  Required  the 
numbers.  Ans.  1,  5,  9,  13. 

7.  A  and  B,  165  miles  distant  from  each  other,  set  out  with  a 
design  to  meet ;  A  travels  1  mile  the  first  day,  2  the  second,  and 
so  on ;  B  travels  20  miles  the  first  day,  18  the  second,  and  so  on. 
How  soon  will  they  meet  ? 

Ans.  They  will  meet  in  10,  and  also  in  33  days. 

8.  The  sum  of  the  squares  of  the  extremes  of  four  numbers  in 
arithmetical  progression  is  200,  and  the  sum  of  the  squares  of  the 
means  is  136.     What  are  the  numbers  ? 

Ans.  14,  10,  6,  2,  or  —  14,  —  10,  —  6,  —  2. 

9.  A  regiment  of  men  was  just  sufiicient  to  form  an  equilateral 
wedge.  It  was  afterwards  doubled,  but  was  still  found  to  want 
385  men  to  complete  a  square  containing  5  more  men  in  a  side, 
than  in  a  side  of  the  wedge.  How  many  did  the  regiment  at 
first  contain  ?  Ans.  820. 

PROGRESSION    BY    QUOTIENT. 

174.  A  series  of  quantities  such,  that  if  any  term  be  divided  by 
the  one  v/hich  precedes  it,  the  quotient  is  the  same  in  whatever 
part  of  the  series  the  two  terms  are  taken,  is  called  a  geometricai 
progression  or  progression  by  quotient. 

The  constant  quotient  is  called  the  ratio  of  the  progression, 


PROGRESSION   BY    QUOTIENT.  iflf 

If  the  series  is  increasing,  the  ratio  will  be  greater  than  unity ; 
if  decreasing,  the  ratio  will  be  less  than  unity. 

The  following  series  are  examples  of  this  kind  of  progres- 
sion, 3  .    6  .  12  .  24  .  48  .  96 
64  .  16  .    4  .    1  .    J  .  tV 

In  the  first  the  ratio  is  2,  in  the  second  J.  A  progression  by 
quotient,  it  will  readily  be  perceived,  is  simply  a  series  of  equal 
ratios  by  quotient,  in  which  each  term  is  at  once  antecedent  and 
consequent,  ivith  the  exception  of  the  first,  which  is  only  an  ante- 
cedent, and  of  the  last,  which  is  only  a  consequent. 

To  indicate  that  the  quantities  a,b,c,d...  form  a  progres- 
sion by  quotient,  we  write  them  thus 

^  a'.h  '.  c  '.  d'. . . ». 

The  progression  is  enunciated  thus,  a  to  ^  as  i  to  c  as  c,  &c. 

175.     Let  us  take  the  general  progression 
^  a'.b:c:d  . . . , 
and  let  the  ratio  be  represented  by  q;  from  the  nature  of  the 
progression,  we  have,  it  is  evident, 

b=iaq,  c=bq^=-aq^,  d  =  cq-=a(f 

from  which  it  will  be  readily  inferred,  that  a  term  of  any  rank 
whatever  is  equal  to  the  first  term  multiplied  by  the  ratio  raised 
to  a  power,  the  exponent  of  luhich  is  one  less  than  the  number, 
which  marks  the  place  of  this  term. 

Let  L  designate  any  term  whate^^er  of  the  progression,  and 
let  n  represent  the  number  of  this  term;  from  what  has  been 
said,  we  have 

This  is  called  the  general  term  of  the  progression.  By  means 
of  it  we  may  find  any  term  required,  when  the  first  term  and  the 
ratio  are  given. 

Thus  let  it  be  required  to  find  the  8th  term  of  the  progression 
•fr  2  :  6  :  18  .  .  .  in  this  case,  we  have 

i  =  2x3'  =  4374. 


218  ELEMENTS   OF    ALGEBRA. 

In  like  manner  if  it  be  required  to  find  the  12th  term  of  the 
progression  -ff  64  :  16  :  4  :  1 :  J  .  .,  we  have 


^  =  «K5)"  =  6i6- 


176  Resuming  the  general  progression 

we  have  from  the  nature  of  the  progression  the  series  of  equa- 
tions, 

b  =  aq,  c  =  hq,  d  =  cq  .  .  .  l:=kq; 
adding  these  equations  member  to  member,  we  have 

b-\-c-{-d-\-.  ,  .l=[a-^rh  +  c-\-.  ,  .k)q.     (1) 
Let  S  represent  the  sum  of  all  the  terms,  we  have 

*  +  c  +  ^  + l=zS  —  a 

a-\-b'-\-  c-\-  .  .  .  .  k=  S  —  I; 
whence  by  substitution  in  equation  (1),  we  have 
S  —  a  =  q{S  —  l), 

and  by  consequence  S  = =-. 

By  means  of  this  formula  we  may  obtain  the  sum  of  any  num- 
ber of  the  terms  of  a  progression  by  quotient ;  for  this  purpose,  we 
multiply  the  last  term  by  the  ratio,  subtract  the  first  term  from 
this  product,  and  divide  the  remainder  by  the  ratio  diminished  by 
unity. 

Let  it  be  required  to  find  the  sum  of  the  first  4  terms  of  the 
progression  -tt  2  :  6  :  IS  :  54  :  162,  we  have 

o  —  1 

When  the  progression  is  decreasing,  that  is,  when  q  is  less 
than  1,  it  will  be  more  convenient  to  put  the  above  expression 

for  S  under  the  form  <S  = ;  since  in  this  case  the  two 

l  —  q 

terms  of  the  fraction  will  be  positive. 

Let  it  be  required  to  find  the  sum  of  the  12  first  terms  of  the 

progression^64:16:4:l:l....^. 


PROGRESSION   BY   QUOTIENT.  019 


64 


,,r    u        o       «  — ^y                65536*4       ^^  ,    65525 
WehaveS  =  3— ^==_— 3; =  85  +  — ^^g. 

4 
177.   If  in  the  formulas  for  S  we  substitute  for  I  its  value, 
viz.  Z  =  ag'''~\  we  have 

aq^'  —  a  a  —  af 

formulas,  by  means  of  which  we  obtain  the  sum  of  any  number 
of  terms  of  a  progression,  when  the  number  of  terms,  the  first 
term  and  ratio  are  given. 

Thus  to  find  the  sum  of  the  first  8  terms  of  the  progression 
rr  2  :  6  :  18  :  54 we  have 

S=-— P  =  -3--^=6560. 

In  the  same  manner,  wc  have  for  the  sum  of  the  12  first  terms 
of  the  progression  -rr  64  :  16  :  4  .  .  . 

65535 


a^aqn       64-64^)^^ 
S  =  -.j-— =  — j— =  85  + 


q  1  '    1966081 

4 


EXAMPLES. 

1.  The  first  term  of  a  progression  by  quotient  is  4,  the  ratio 
3,  and  the  last  term  78372.     What  is  the  sum  of  all  the  terms  ? 

2.  The  first  term  of  a  progression  by  quotient  is  8,  the  last  term 

ncxAci^  ^^^^  ^^®  r^^io  9'     What  is  the  sum  of  the  progression  ? 

3.  The  first  term  of  a  progression  by  quotient  is  3,  the  ratio 

7 

-,  and  the  number  of  terms  10.     Required  the  sum  of  the  pro- 
gression. 

4.  The  first  term  of  a  progression  by  quotient  is  -,  the  ratio 

2  ' 

s,  and  the  number  of  terms  9.     What  is  the  sum  of  the  pro- 

gression  ? 


220  ELEMENTS    OF    ALGEBRA. 

INFINITE    PROGRESSIONS    BY    QUOTIENT. 

178.  Let  there  be  the  decreasing  progression 
ir  (t-  b  '  c:  d  .  .  , 
consisting  of  an  infinite  number  of  terms.     The  formula  for  the 

sum  of  any  number  of  terms,  viz.    S  =  — may  be  put 

under  the  form 

But  since  the  progression  is  decreasing,  gf  is  a  fraction ;  ^  is 
also  a  fraction ;  hence  as  the  number  n  becomes  greater,  or  as 

we  take  more  terms,  the  expression  -z .  9"  becomes  smaller, 

and  the  value  of  S  approaches  nearer  to  -z .     If  then  we 

suppose   n  greater    than    any   assignable   quantity   or  infinite, 
.  g-"  will  be  less  than  any  assignable  quantity  or  0,  and 

will  in  this  case  represent  the  true  value  of  the  series. 


\-q 

a 


l-q 

We  conclude,  therefore,  that  the  sum  of  the  terms  of  a  de- 
creasing progression,  in  which  the  number  of  terms  is  infinite, 

has  for  its  expression  S  = ,  q  being  the  ratio  of  the  pro- 
gression and  a  the  first  term. 

179.  Strictly  speaking  the  quantity is  the  limit  which 

the  sum  of  a  decreasing  progression  can  never  surpass,  but  to 
which  it  continually  approximates  as  we  take  more  terms. 
Let  there  be,  for  example,  the  progression 

we  have  a=\,  q=\,  whence 

e__     O'  a        „_     2  2  /iy_     2 ]_ 

— 1__^      l_g-^— SZTi— 2iriXV2/  ~2— 1      2"-»' 
Here  the  greater  the  value  of  n  or  the  more  terms  we  take, 

the  less  is  the  fraction  ^^^zrv  ^^^  the  nearer  the  sum  of  the  series 


PROGRESSION    BY    QUOTIENT.  ,  221 

approaches  to  2.     If  the  number  of  terms  be  considered  infinite, 
the  fraction  ^^^—^  will  be  less  than  any  assignable  quantity  or 

zero,  and  the  sum  of  the  series  will  be  equal  to  2. 

Strictly  speaking,  however,  2  is  the  limit,  which  the  sum  of 
the  proposed  series  can  never  surpass,  but  to  which  it  constantly 
approximates  as  we  take  more  terms. 

Thus  let  the  nimiber  of  terms  be  1,  2,  3,  4  .  .  .  .  succes- 
sively, we  have 


1 

=  2—1 

1+i 

=  2-4 

1+i+i 

=  2-J 

l+J+i+4 

=  2-4 

i  +  4  +  J  +  4  +  tV=2-A- 

Here  the  more  terms  we  take,  the  nearer  the  sum  of  the  pro- 
gression will  approach  to  2,  from  which  it  may  be  made  to  differ 
by  a  quantity  as  small  as  we  please,  though  strictly  speaking,  it 
can  never  become  equal  to  2. 

180.  When  the  series  is  increasing,  that  is,  when  q  is  greater 

than   unity,   the    expression    S  = cannot  be   considered 

l  —  q 

as  the  limit,  which  the  s\im  of  the  series  can  never  surpass. 

For    the    sum    of   a    determinate    number   n    of    terms    being 

o           «             «?"       .     .         .,  ,  a(f        M,    . 

o== ,   It   IS   evident,   that   :; — - —   will  mcrease 

l_gr  l  —  q  l_gr 

more  and  more  numerically  in  proportion  as  n  increases;  by 
consequence  the  more  terms  we  take,  the  more  will  the  sum  of 

the  terms  differ  numerically  from  :; .     In  this  case is 

l  —  q  l  —  q 

merely  the  algebraic  expression,  which  by  its  development  gives 

rise  to  the  series 

a-{'aq-\'aq^-{-  ag^-\-  .  .  .  . 

Indeed  if  we  perform  upon  a  the  division  indicated,  we  have 

Yzr^=(^  +  a9  +  af  +  a^+ 


223  •  ELEMENTS    OF   ALGEBRA. 

181.  In   the   above   expression   let  a=l,  g'  =  2,  we   have 

_1_  or- 1  =  1  +  2  +  4  +  8+16+... 

an  equation,  in  which  the  first  member  is  negative,  while  the 
second  is  positive,  and  greater  in  proportion  as  q  is  greater. 

In  order  to  interpret  this  result  we  observe,  that  if  in  the  equa- 
tion  =  a-\-  aq-\-  agf^  .  .  .  .  we  stop  the  series  at  any  par- 
ticular term,  it  is  necessary,  in  order  to  preserve  the  equality  of 
the  two  members,  to  complete  the  quotient  by  annexing  to  it  the 
fraction  which  remains.  If,  for  example,  we  stop  the  series  at 
the  fourth  term  a  cf,  we  shall  have  by  completing  the  quotient 

——=a  +  aq-\-aq^-}-a(f  +  Yir^' 
an  equation  which  is  exact.     Indeed   if  in  this   equation,  we 
make  a=l,  q  =  2,  we  have 

-l  =  l  +  2  +  4  +  8  +  i^; 

from  which  we  obtain  —  1  =  —  1. 

EXAMPLES. 

1.  What  is  the  sum  of  the  infinite  progression 

1  :  1  :  i  :  2-V  :  .  .  .  . 

2.  What  is  the  sum  of  the  progression  2  :  f  :  f  : con- 
tinued to  infinity  ? 

3.  The  first  term  of  a  geometrical  progression  is  a,  and  the 

ratio .     What  is  the  sum  of  this  progression  continued  to 

infinity  ? 

4.  What  vulgar  fraction  is  equivalent  to  the  repeating  deci- 
mal 3  ? 

This  decimal  may  be  put  under  the  form 

^  (tV  +  T^-^  +  TTjW  +  TIT^XFTF  + ) 

5.  What  vulgar  fraction  is  equal  to  the  repeating  decimal  25 ; 
what  to  the  decimal  375  ? 

182.  The  equations  lz=agr-\  S=  ^      ,    contain  all  the 

*  ff_i 


J?BOGRESSION    BY    QUOTIENT. 


tm 


relations  of  the  five  quantities  a,  I,  q,  n,  and  S;  we  have  then 
the  general  problem,  any  three  of  the  Jive  quantities,  a,  1,  q,  n  and 
S  being  given  to  find  the  remaining  two.  This  general  problem 
gives  rise  to  ten  particular  problems,  the  enunciations  of  which 
will  not  differ  from  those  relative  to  progressions  by  difference, 
art.  171,  with  the  exception,  that  the  ratio  is  here  expressed  by 
the  letter  q  instead  of  d. 

183.  From  the  formula  Z  =  ag'"~S  we  obtain 


-v/^- 


This  expression  for  q  enables  us  to  resolve  the  following 
problem,  viz.  to  insert  between  two  given  numbers,  b  and  c,  m 
mean  proportionals,  that  is  to  say,  a  number  m  of  quantities, 
which  comprised  between  b  and  c  will  form  ivith  them  a  progreS' 
sion  by  quotient. 

To  resolve  this  problem  it  will  be  sufficient  to  determine  the 
ratio  of  the  progression  required ;  for  this  we  have  given  the  first 
term  b,  the  last  term  c  and  the  number  of  terms  m  -j-  2. 

Substituting  therefore  b,  c  and  m-\-2  ioT  a,  I  and  n  in  the 
above  expression  for  q,  we  have  for  the  ratio  of  the  required 
progression 

whence  to  find  the  ratio  sought,  ive  divide  the  given  numbers  b 
and  c,  one  by  the  other,  and  extract  the  root  of  the  quotient  to  the 
degree  marked  by  the  number  of  terms  to  be  i?iserted  plus  one. 

Let  it  be  required  to  insert  six  mean  proportionals  between  the 
numbers  3  and  384.     Here  ?/z  =  6,  we  have  therefore 


.  =  ^f  =  ^Ils  =  . 


The  progression  required  is  therefore 

4f  3  :  6  :  12  :  24  :  48  :  96  :  192  :  384. 
From  what  has  been  done,  it  will  be  easy  to  see,  that  if  between 
the  terms  of  a  progression  by  quotient  taken  two  and  two,  we  in' 


224  ELEMENTS    OF    ALGEBRA. 

sert  the  same  number  of  mean  proportionals,  the  partial  progres* 
sions  thus  formed  will  together  form  a  progression  by  quotient. 

1S4.  Of  the  ten  particular  problems,  which  may  be  proposed 
upon  progressions  by  quotient,  four  only  can  be  resolved  by 
principles  thus  far  laid  down.  Below  we  have  the  enunciation 
of  these  problems  with  their  answers. 

1°.  a,  q,  n  being  given  to  find  I  and  S. 

a{q^-\) 


l-=.a^ 


— 1 


1  n-l 


S— 1 

2°.  a,n,l  being  given  to  find  q  and  S. 

n_l 

3**.  q,  n,  I  being  given  to  find  a  and  S. 


a=  -—7,    o  = 


4®.  q,  n,  S  being  given  to  find  c  and  I. 

S(?-l)     ^       Sg-'(g-l) 


Of  the  remaining  problems,  two,  viz.  those  in  which  a  and  q, 
I  and  q  are  the  unknown  quantities,  depend  upon  the  resolution 
of  equations  of  a  degree  superior  to  the  second.  The  other  four 
depend  upon  the  resolution  of  an  equation  of  a  nature  altogether 
different  from  any  which  we  have  yet  seen,  viz.  upon  an  equa- 
tion of  the  form  c*  =  &  in  which  the  exponent  is  the  unknown 
quantity. 

QUESTIONS  PRODUCING  PROGRESSIONS  BY  QUOTIENT. 

1.  There  are  three  numbers  in  geometrical  progression,  the 
greatest  of  which  exceeds  the  least  by  15.  Also  the  difference  of 
the  squares  of  the  greatest  and  least  is  to  the  sum  of  the  squares 
of  all  the  three  numbers  as  5  to  7.     Eequired  the  numbers. 

Let  X,  xy,  xy^  be  the  numbers ;  then  by  the  question  we 
have  xif  —  a:  =15, 

and  7  {x'y*  ^a^)  =  5  {x'f  +  a:V  +  :c»), 

or  by  division        7{y*  —  1)  =  5  (i/*  +  ^  +  1). 


PROGRESSION  BY  QUOTIENT.  225 

or  performing  the  operations  indicated,  transposing  and  reducing 

whence,  resolving  this  last,  we  have 

^  =  4,  and  2/  =  2. 
Substituting  next  for  y  its  value  in  the  first  equation,  we  ol 
tain  a:  =  5.     The  numbers  required  are  therefore  5,  10  and  2f  •'. 

2.  The  sum  of  three  numbers  in  geometrical  progression  is  13, 
and  the  product  of  the  mean,  and  the  sum  of  the  extremes  is  30. 
Required  the  numbers. 

X 

Let  the  numbers  be  -,  x  and  xy\  then  by  the  question,  we 
have 

J  +  ^  +  ^y  =  13. 

and  i--\-xy\x  =  ^0. 

By  transposition  in  the  first  equation,  we  have 
— \-xy=.\^  —  X  ; 

y 

whence,  by  substitution  in  the  second,  we  obtain 

{n  —  x)x  =  '^0; 
whence  a?  —  13  z  =  —  30, 

from  which  we  deduce 

x=\0,x  =  2. 
Substituting  the  value  a:  =  3  in  the  first  equation,  we  obtain 

y  =  3,  or  jr,  and  the  numbers  sought  are  1,  3,  9. 

3.  The  difference  between  the  first  and  second  of  four  num- 
bers in  geometrical  progression  is  36,  and  the  difference  between 
the  third  and  fourth  is  4.     What  are  the  numbers  ? 

Ans.  54,  18,  6,  and  2. 

4.  A  gentleman  divided  £210  among  three  servants  in  geo- 
metrical progression;  the  first  had  £90  more  than  the  last. 
How  much  had  each  ? 

5.  There  are  three  numbers  in  geometrical  progression,  the 

15 


226  ELEMENTS   OF   ALGEBRA. 

sum  of  the  first  and  second  of  which  is  9,  and  the  sum  of  the 
first  and  third  is  15.     Required  the  numbers. 

6.  The  sum  of  three  numbers  in  geometrical  progression  is 
35,  and  the  mean  term  is  to  the  difference  of  the  extremes  as 
2  to  3.     Required  the  numbers.  Ans.  5,  10,  20. 

7.  The  sum  of  £  14  was  divided  between  three  persons,  whose 
shares  were  in  geometrical  progression ;  the  sum  of  the  shares 
of  the  first  and  second  was  to  the  sum  of  the  shares  of  the 
second  and  third  as  1  to  2.     Required  the  shares. 

Ans.  2,  4,  8. 


SECTION  XXII.— Theory  of  Continued  Fractions. 

185.  In  order  to  form  a  more  exact  idea  of  a  fraction,  the 

terms  of  which  are  large  numbers  and  prime  to  each  other,  we 

seek  approximate  values  of  this  fraction,  which  are  expressed  in 

more  simple  numbers. 

159 
Let  there  be,  for  example,  the  fraction  j^.     Dividing  both 

terms  of  this  fraction  by  the  numerator,  an  operation  which  will 

not  change  its  value,  it  becomes r^ 

Ifi 

If  then  we  neglect,  for  the  moment,  the  fraction  r-^  in  this  ex- 

lo9 

pression,  the  result  -  will  be  greater  than  the  proposed,  since  the 
o 

denominator  has  been  diminished. 

Ifi 
On  the  other  hand,  if  instead  of  neglecting  the  fraction  r-^, 

we  substitute  1  for  it,  the  result  -  will  be  less  than  the  proposed, 
since  the  denominator  has  been  increased. 

We  conclude  therefore,  that  the  fraction  ^qq  is  comprised  be- 
tween ^  and  2>  we  are  thus  enabled  to  form  a  very  exact  idea  of 
its  value. 


CONTINUED   FRACTIOT^S.  227 

If  a  greater  degree  of  approximation  be  required,  we  have 

only  to  operate  upon  r-r^  m  the  same  manner  as  we  have  al- 

159 
ready  done  upon  ^^;  we  have  thus 

.16        1 


159       ,.   .   15 


^  +  16 


and  the  proposed  fraction  becomes 
1 

3  +  ^ 


9-4-^. 
^+16 

If  we  ne2:lect  -^^^  ^  is  greater  than  7777 ;  it  follows  therefore 
lb    y  lo9 

^       1  .    ,         ^       159     ^       1  ^  1  9 

that  7  IS  less  than  77^;   but becomes  kf,  or  ^rp,; 

^  ,    1  493'  1  28        28' 

^+9  ^  +  9  -9 

1  9 

thus  the  proposed  is  comprised  between  -  and  tr^.     The  differ- 

ence  between  these  two  fractions  is  pr7 ;  the  error  therefore  com- 

84 

19 

mitted  in  taking  k  or  rr^  for  the  value  of  the  proposed  fraction  is 


less  than  — . 

To  attain  to  { 

I  still 

[  greater  degree 

of  J 

ipproximation, 

we 

1 
operate 

in  the  same  manner 

15 

upon  -; 

thus 

we 

have 

* 

15 
16  ~ 

1 
1  + 

T 

15 

and  the  proposed  fraction  becomes 

1 

«  .   1 


228  ELEMENTS   OF    ALGEBRA. 

11  15 

Neglecting  --p-  the  fraction  -  or  1  is  greater  than  ~ ;  hence 
xOf  1  lb 

7  or  VFJ  is  less   than  —^ ;    therefore  ~ 

,  +  '        ">  »»»  3  +  1 


or  KT  is  greater  than  -r^;  thus  the  proposed  is  comprised  be» 

9  10 

tween  ^r^  and  -^,     The  difference  between  these  two-  fractions  m 
Jo  ol 

1        1.  .     1     ......  ^       9         10 

5^3-;  the  error  committed,  therefore,  m  takmg  either  ^^  01  -^ 

for  the  value  of  the  proposed  is  less  than  3-3. 

000 

The  expression is  called  a  continiied  frac' 


=+; 


'+A 


tvon.  We  understand  therefore,  by  a  continued  fraction  a  frac- 
tion^ which  has  unity  for  its  numerator,  and  for  its  denominator 
an  entire  number  plus  a  fraction,  which  fraction  has  also  unity 
for  its  nuTnerator  and  for  its  denoTmnutor  an  entire  naimher  plus 
a  fraction,  and  thus  in  order. 

It  sometimes  happens,  that  the  proposed  fractional  number  is 
greater  than  unity ;  to  generalize,  therefore,  the  above  definition,, 
we  say,  that  a  continued  fraction  is  an  expression  composed  of  an 
entire  number  plus  a  fraction  which  has  unity  for  its  numerator t 
and  for  its  denominat&r  an  entire  number  plus  a  fraction,  Sf&, 

159 
186.  If  we  examine  the  above  process  for  converting  j^  into 

a  continu-ed  fraction,  it  will  be  perceived,  that  we  have  divided 
first  493  by  159,  which  gives  three  for  a  quotient  and  a  remain- 
der 16;  we  then  divide  159  by  16,  which  gives  9  for  a  quotient 
and  a  remainder  15 ;  we  next  divide  16  by  15,  which  gives  1  for 
a  quotient  and  a  remainder  1 ;  from  which  we  readily  infer  the 
following  rule  for  converting  a  fraction  or  fractional  number  into 
a  continued  fraction,  viz. 


CONTINUED  FRACTIONS. 

Apply  to  the  two  terms  of  the  fraction  the  process  of  finding 
fheir  greatest  comrmn  divisor ;  pursue  the  operation  until  a  re- 
mainder is  obtained  equal  to  0 ;  the  successive  qicotients,  thus  ob- 
tained, will  he  the  denominators  of  the  fractions,  which  constitute 
the  continued  fraction. 

If  the  proposed  be  greater  than  unity  the  first  quotient  will  be 
the  entire  part,  which  enters  into  the  expression  of  the  continued 

fraction. 

73    829 

Examples.  Let  the  fractions  r-r-,  jr^  be  converted  into  con- 
tinued fractions. 

187.  From  what  has  been  said  a  continued  fraction  may  be 
represented  generally  by  the  expression 


c+^ 


a,  b,  c,  d  ,  ,  ,  being  entire  and  positive  numbers.  The  fractional 
number,  to  which  this  expression  is  equivalent,  may  moreover  be 
represented  by  x. 

The  fractions  y,  -,-...,  the  assemblage  of  which  constitutes 
oca 

the  continued  fraction,  are  called  integrant  fractions.  The  de- 
nominators b,  c,'d . .. .  are  called  incomplete  quotients,  since  b, 
for  example,  is  only  the  entire  part  of  the  number  expressed 

hy  b-\ r  and  c  only  the  entire  part  of  the  number  ex- 

^  +  1^7. 

pressed  by  c  -f-  j-j- —  and  thus  in  order.     Conversely  the  ex- 

1  1 

pressions  b  -] =■  c  -}-  -i —  are  called  complete  quotients. 

+  1  a , . 

n 

The  results  obtained  by  converting  successively  into  a  single 
fractional  number  each  of  the  expressions 

a  -j-  T)  ^-\ T  &c«  are  called  reductions. 

rp  t: 


230  ELEMENTS    OF    ALGEBRA. 

188.   The  formation  of  these  reductions  is  subject  to  a  very 
simple  law,  which  we  shall  now  develop. 

The  first  is  a,  which  may  be  put  under  the  form  y,  the  second 

is  a  4-  -T)  or  reducing  the  whole  expression  to  a  fraction,  — y-—' 

To  form  the  third,  represented  by 

.    1 


'+i 


c  -|-  -T  for  c  in  the  third ;  which  gives 


it  will  be  sufficient  to  substitute  h  -\ —  for  h  in  the  second ; 

c 
.  *      . 
making  this  substitution,  we  have 

'    c  '    c 

To  form  the  fourth  reduction,  it  will  be  sufficient  to  substitute 

J  third ;  which  gives 

[{ab  +  \)c-\-a]d  +  ab+l 
{bc  +  \)d  +  b 

The  first  four  reductions  therefore  will  be 

a      ab-\-l      {ab-\-\)c-\-a      {{ab -\- \)  c -[- a]  d -{- ab -{- I 
V  b      '  bc+\        '  {bc+l)d-\-b 

Without  proceeding  further,  it  will  be  perceived,  that  the 
numerator  of  the  third  reduction  is  formed  by»  multiplying  the 
numerator  of  the  second  by  the  third  incomplete  quotient  c, 
and  adding  to  this  product  the  numerator  of  the  first  reduction. 
With  respect  to  the  denominator,  it  is  formed  in  the  same 
manner  by  means  of  the  denominators  of  the  second  and  first 
reductions. 

In  like  manner,  the  numerator  and  denominator  of  the  fourth 
reduction  is  formed,  it  will  be  perceived,  by  multiplying  re- 
spectively the  two  terms  of  the  third  reduction  by  the  fourth 


JO 

form  the  reduction  ^^,  and  let  it  be  supposed,  that  we  have 
Jx 


CONTINUED   FEACTIONS.  SSI 

incomplete  quotient  d  and  adding  to  the  two  products  respectively 
the  two  terms  of  the  second  reduction. 

From  what  has  been  done  it  will  be  readily  inferred,  that  the 
above  law  of  formation  for  the  third  and  fourth  reductions  should 
be  extended  to  those  which  follow.  To  demonstrate  this  law, 
however,  in  a  rigorous  manner,  we  shall  show  that  if  it  be  true 
in  regard  to  any  three  successive  reductions  whatever,  it  will  be 
true  for  the  reduction,  which  follows ;  thus  this  law  being  already 
found  true  for  the  first  three  reductions  will  be  true  for  the 
fourth,  and  being  true  for  the  second,  third  and  fourth,  it  will 
be  true  for  the  fifth,  and  thus  in  order;  it  will  therefore  be 
general. 

POT? 

Let  ^„  7r-„  :^  be  any  three  successive  reductions  whatever,* 

let  r  be  the  incomplete  quotient,  at  which  we  stop  in  order  to 
,  and  let  it  be 

R  _  Qr  +  P 
R'~~  Q'r  +  P'' 

Let  -  be  the  integrant  fraction,  which  follows  r,  and  let  ~^, 

S  O 

be  the  corresponding  reduction.  In  order  to  form  this  reduc 
tion,  it  is  sufficient  to  substitute  in  the  expression  for  -^,  r  -{-- 
instead  of  r  ;  making  this  substitution,  we  have 

S  _^y+'^)  +  ^  _{'Qr+  P)s+Q  _Rs+Q 

S 
We  see,  therefore,  that  —  is  formed  from  the  two  preceding 
o 

reductions  according  to  the  law  enunciated  above.  This  law  is 
therefore  general ;  whence,  To  form  the  numerator  of  any  re- 
duction whatever,  we  multiply  the  numerator  of  the  preceding  re- 
duction  hy  the  incomplete  quotient,  which  corresponds  to  it,  and 
add  to  the  product  the  numerator  of  the  reduction,  which  precedes 
hy  tivo  ranks  the  one  which  we  wish  to  form  ;  the  denominator  is 


ELEMENTS    OF    ALGEBRA. 

formed  by  the  same  law  by  means  of  th£  two  'preceding  denomina- 
tors. 

189.  When  the  number  reduced  to  a  continued  fraction  is 

less  than  unity,  we  substitute  y  instead  of  a,  in  order  to  apply  the 

law,  which  supposes  necessarily,  that  we  have  already  the  first 
two  reductions. 

Let  it  be  proposed  to  find  the  successive  reductions  of  the 
continued  fraction 

65_0       1 

1  ~r 


149       1   '        .   1 


2  +  i 


1+i 


The  first  two  reductions  being  - ,  -,  we  have  for  those  which 

J.        /i 

follow 

3    7     17   24    65 

7'  16'  39'  55'  149* 

In  like  manner  we  have  for  the  several  reductions  of  the 

continued  fraction  arisinf?  from  ^r— -, 
°  347 

2    5   7    12    43    829 

I'  2'  3'  T'  18'  347* 

29 
So  also  the  fraction  —  being  converted  into  a  continued  frac- 
tion gives  the  following  reductions,  viz.  ^ 
0    1    1    2   3   29 
1'  2'  3'  5'  8'  77* 
190.  The  successive  reductions,  it  will  be  perceived,  are  alter- 
nately less  and  greater  than  the  whole  continued  fraction,  and 
they  approximate  this  fraction  nearer  and  nearer. 

The  first  reduction  is  always  less  thah  the  whole  continued 
fraction  x.  The  reductions  of  an  even  rank  are,  therefore,  greater 
than  the  vjhole  continued  fraction,  and  those  of  an  odd  rank  are 
less.     And  since  these  reductions  approach  nearer  and  nearer 


CONTINUED  FRACTIONS.  8iB 

the  value  of  a;,  the  reductions  of  an  odd  rank  must  go  on  increas- 
ing, while  those  of  an  even  rank  decrease.  Thus  the  reductions 
form  two  series,  the  terms  of  which  approach  nearer  and  nearer 
the  value  of  the  whole  continued  fraction. 

191.  The  difference  between  any  two  consecutive  reductions 
whatever  has  unity  for  its  numerator.  The  numerator  of  the 
first  difference  is  -j-  1,  that  of  the  second  — 1,  that  of  the  third 
-f- 1,  and  thus  in  order.  In  general,  the  numerator  of  any  differ- 
en£e  whatever  will  he  +1,  if  the  second  of  the  reductions  under 
consideration  is  of  an  even  rank,  but  —  1  if  it  be  of  an  odd  rank. 

From  this  property,  it  follows,  that  the  two  terms  of  any 

72 

reduction  whatever  ^7  are  prime  to  each  other. 

Indeed  let  it  be  supposed,  that  R  and  R'  have  a  common 
factor  h;  by  the  preceding  property,  we  have 

whence  dividing  both  terms  by  h,  we  have 

RQ'      QR'__l, 

h  h         h 

but  the  first  member  of  this  equation  is  an  entire  number  since 
by  hypothesis  R  and  K  are  divisible  by  h,  while  the  second  is 
essentially  a  fraction;  jR  and  R'  cannot  therefore  have  a  com- 
mon factor. 

From  this  it  follows,  that  if  a  fraction,  the  terms  of  which  are 
not  prime  to  each  other,  be  converted  into  a  continued  fraction, 
and  all  the  reductions  be  formed  to  the  last  inclusive,  the  last 
reduction  will  not  be  the  proposed  fraction,  but  this  fraction 
reduced  to  its  lowest  terms. 

348 
Let  there  be,  for  example,  the  fraction  ^^;  converting  this 

into  a  continued  fraction,  we  have  for  the   successive  reduc- 

0    1    1    2   3   29  29 

tions  -,  ^,  35,  ^,  ^,  ==.     The  last  reduction  --  is  the  proposed 

reduced  to  its  lowest  terms. 

192.  Since  the  value  of  the  whole  continued  fraction  x  is 


ELEMENTS   OF   ALGEBRA. 

always    comprised    between    any   two    consecutive    reductions 
•p:rj,  :g7,  it  follows  that  the  error  committed  in  taking  —,.  for  z  is 

O        TJ. 

less  than  -z^  —  ^^;  but  from  what  has  been  said  we  have 

Q       R  1 


Q'      R'       Q'R' 
and  since  Q'<R'  gives  Q"^<  Q'R',  we  have 


Q'R'  ^Q' 

The  error  therefore  committed  in  taking  any  reduction  what- 
ever for  the  value  of  the  whole  continued  fraction  is  less  than 
unity  divided  by  the  denominator  of  this  reduction  multiplied  by 
the  dcTwrninator  of  the  reduction  ivhich  folloios,  or  less  exactly 
but  in  terms  more  simple,  less  than  unity  divided  by  the  square 
of  the  denominator  of  the  reduction,  which  is  taken  for  the  whole 
continued  fraction. 

193.    The  ratio  of  the   circumference   to  the  diameter  of  a 

circle  being  expressed  by  the  fraction  fH|.  the  terms  of 

which  are  prime  to  each  other,  let  it  be  proposed  to  find  a 

fraction,  the  terms  of  which  will  be  more  simple,  and  which 

will  express  the  same  ratio  nearly.     Converting  the  proposed 

into  a  continued  fraction,  we  have  for  the  successive  reductions 

3   22   333   855   9208   9563   76149   314159 

r  y  106'  113'  2931'  3044'  24239'  100000* 

The  error  committed  in  taking  the  second  of  these  reductions 

1       22 

for  the  proposed  fraction  will  not  exceed  -^;  ■—  is  therefore  fre- 
quently employed  to  express  the  ratio  of  the  circumference  of  a 
circle  to  its  diameter.     This  is  the  ratio  given  by  Archimedes. 

If  a  greater  degree  of  approximation  is  required,  we  take 
the  fourth  reduction,  which  it  is  easy  to  see,  is  but  little  more 
complicated  than  the  third.     The  error  committed  in  taking  this 

1  855 

reduction  for  the  proposed  will  not  exceed  iio  ^  t)Qoi  5  TTo  will 


EXPONENTIAL   EQUATIONS. 

therefore  approximate  the  proposed  very  nearly.     This  is  the 
ratio  given  by  Adrian  Metiiis. 

We  thus  see  the  use,  which  may  be  made  of  continued  frac- 
tions in  estimating  approximatively  the  value  of  fractions,  the 
terms  of  which  are  large  numbers  and  prime  to  each  other. 

EXAMPLES. 

1.  A's  property  is  to  that  of  B  as  5743  to  80937.  By  what 
smaller  numbers  ma^  the  ratio  of  their  property  be  expressed  ? 

2.  The  lunar  month  or  the  time  in  which  the  moon  completes 
its  revolution,  is  found  by  calculation  to  be  27.321661  days. 
Thus  in  27321661  days  it  would  perform  1000000  revolutions. 
How  may  this  relation  be  expressed  in  smaller  numbers  ? 


SECTION  XXIII. — Exponential  Equations  and  Logarithms. 

194.  An  equation  of  the  form  a' =  3,  in  which  the  exponent 
z  is  the  unknown  quantity  is  called  an  exponential  equation. 
The  solution  of  this  equation  consists  in  finding  the  power,  to 
which  it  is  necessary  to  raise  a  given  quantity  a  in  order  to  pro- 
duce another  given  quantity  h. 

Let  there  be,  for  example,  the  equation  2""  ==  64 ;  raising  2  to 
its  different  powers,  we  soon  find  that  2^  =  64 ;  a:  =  6  answers 
therefore  the  conditions  of  the  equation. 

Again,  let  there  be  the  equation  3''  =  243;  raising  3  to  its 
different  powers  we  find  3^  =  243 ;  whence  a;  =  5.  In  a  word, 
so  long  as  the  second  member  i  is  a  perfect  power  of  the  given 
number  a,  x  will  be  an  entire  number  and  its  value  may  be  found 
by  raising  a  successively  to  its  different  powers,  beginning  with 
that,  the  exponent  of  which  is  0. 

Let  it  be  proposed  next  to  resolve  the  equation  2'  =  6. 
Putting  successively  a;  =  2,  a;  =  3,  we  have  2^  =  4,  2^  =  8; 
the  value  of  x  is,  therefore,  comprised  between  the  numbers  2 
and  3. 


33d  elements  of  algebra. 

Let  us  put  therefore,  x  =  2-\-—,  x'  being  greater  than  1  ; 

substituting  this  value  in  the  proposed,  we  have 

2  +  ^  1  1-         Q 

2     "  =  6  or  2=^  X  2"'  =  6,  whence  2''  =  1 
or  raising  both  members  to  the  power  x\  we  have 

To  determine  the  value  of  x'  we  make  successively  x'  •=.\^ 

(3\         3  /3\*       9 

-  J   or  ^  less  than  2,  but  I- j   or  j 

greater  than  2 ;  a:'  is,  therefore,  comprised  between  1  and  2. 

Let  us  put  then  a;'=  1  +  -77,  x"  being  greater  than  1.     Sub- 
stituting this  value,  we  have 

(|)-i=,„|x(|)»=., 

/4\'"_3 
\?)    "^2 

To  determine  the  value  of  x" ^  we  make  successively  x"  •=.  1, 

^       16 


whence 


(4\         4  3  /4\'' 

jr  I    or  -  less  than  -,  but  1  -  1 

3 

greater  than  - ;  x"  is,  therefore,  comprised  between  1  and  2. 

Let  us  put  then  a;"=  1  -| — —^  x'"  being  greater  than  unity; 

X 

we  have  by  substitution 

Uj         =2  °'  3  X  W     =2'  ^^^^^^  Vsj    =3- 

(Q\ 8        ft! 
ft)  ^^fil'^^^"^' 

4         /9\^      729  4 

ber  less  than  -  but  (  -  I  =  — ^,  a  number  greater  than  -^  \  thus 

x'"  is  comprised  between  2  and  3. 

Let  x'"  =  2  -| — 7777,  the  equation  in  x'"  becomes 

/9\=+b^       4       ,  /256\*""      9 

is;         =3'  ^^^^^H243J     =8- 


KXFONENTIAL   EQUATIONS.  SP^ 

Operating  upon  this  last  equation  as  upon  the  preceding,  we 
find  two  entire  numbers  k  and  ^+1,  between  which  x""  will 

be  comprised.     Putting  x"" z=Jc-\-—,  we  determine  a;*,  in  the 

same  manner  as  we  have  already  done  x"'\  and  thus  in  order. 
Bringing  together  the  equations 

we  obtain  the  value  of  x  under  the  form  of  a  continued  fraction, 
thus 

1+ 


But  we  have  seen  that  in  a  continued  fraction  the  greater  the 
number  of  integrant  fractions,  which  are  taken,  the  nearer  we 
approach  the  value  of  the  number  reduced  to  a  continued  frac- 
tion ;  we  shall,  therefore,  be  able  to  determine  the  value  of  x  in 
the  equation  2'  :^  6,  if  not  exactly,  at  least  with  such  degree  of 
approximation  as  we  please. 

Forming   the   first   four   reductions,   for   example,   we   have 

2   3   5    13         ,    ^        ,      .       13  ,.^      ^ 

T'  T'  9'  "T '  ^  reduction  —  diners  from  a:  by  a  quantity 

less  than  — . 

To  attain  a  greater  degree  of  approximation,  we  determine 

(256\*""      9 
oTq)     =qJ  we  thus  find 

a;""  =— 2  4" -;•    We  shall  have,  then,  for  the  fifth  reduction 

X 

31  1 

•r^.     This  differs  from  a;  by  a  quantity  less  than  r^. 

195.  From  the  preceding  examples  the  course  to  be  pursued 
in  the  solution  of  equations  of  the  form  a*  =  3  will  be  readily 


m" 


ELEMENTS    OF   ALGEBRA. 


inferred.  In  the  application  of  this  method  to  particular  cases  it 
is  necessary  to  remark,  1°.  If  the  quantity  b  be  less  than  a,  the 
value  of  X  will  be  comprised  between  0  and  1 ;  we  put,  therefore, 

x  =  —.  ^°.  If  &  is  a  fraction  and  a  greater  than  unity,  the 
value  of  X  will  be  negative,  we  put,  therefore,  x  =  —  y ;  the 

equation  is  then  reduced  to  the  form  or'z^-;  having  found  the 

value  of  y  in  this  equation  according  to  the  method  explained 
above,  the  value  of  x  will  be  equal  to  that  of  y  taken  negatively. 

i 

EXAMPLES. 

I*'.  Given  3^=  15  to  find  the  value  of  x.      Ans.  x  =  2.46. 

2?.       .    10'==3       .         .        .         .        Ans.  a;  =  0.47. 

2 
3*>.       .      5*=- to  find  the  value  of  a;.    Ans.  a:  =  —  0.25. 


?   .      .      .         .  Ans.  a;  =  0.53. 

4 


In  the  above  examples  the  reductions  furnished  by  the  method 
are  converted  into  decimal  fractions,  and  the  value  of  x  is  deter- 
mined to  within  .01. 

THEORY    OF   LOGARITHMS. 

196.  If  in  the  equation  a"  =  y,  we  assign  a  constant  value 
different  from  unity  to  a,  and  suppose  that  of  z  to  vary,  as  may 
be  required,  we  may  obtain  successively  for  y  all  possible  num- 
bers. 

Let  us  suppose  first  a  greater  than  1. 

If  we  make  successively        a;  =  0,  1,  2,  3,  4,  .  .  .  . 
we  have  y=lf  a,  a^,  a^,  a*,  .  .  .  . 

Thus  by  means  of  the  powers  of  a,  the  exponents  of  which  are 
positive,  entire  or  fractional,  we  may  produce  all  possible  posi^ 
tive  numbers  greater  than  1. 

Again,  let  a;  =  0,  —  1,  — 2,  — 3,  — 4, 

1.  ,1111 

wehav«         j^«l,  -,-,_,  ^,.... 


THEORY   OF   LOGARITHMS. 


m 


Thus  by  means  of  the  powers  of  a,  the  exponents  of  which  are 
negative  entire  or  fractional,  we  may  produce  all  possible  positive 
numbers  less  than  1. 

If  on  the  other  hand  we  suppose  a  less  than  unity,  still  all 
possible  positive  numbers  may  be  produced  by  means  of  the 
different  powers  of  a,  only  the  order  in  which  they  are  produced 
will  be  reversed. 

We  see  therefore,  that  all  possible  positive  iiumbers  may  be 
produced  by  means  of  any  positive  number  whatever  a,  different 
from  unity,  by  raising  this  number  to  the  requisite  powers. 

It  is  necessary,  that  a  should  be  different  from  unity,  other- 
wise the  same  number  will  be  produced,  whatever  value  we 
assign  to  x. 

197.  Let  it  now  be  supposed  that  wfe  have  made  a  table  con- 
taining in  one  column  all  entire  numbers,  and  by  the  side  of 
these  in  another  column  the  exponents  of  the  powers,  to  which  it 
is  necessary  to  raise  a  constant  number  in  order  to  produce  these 
numbers;  this  would  be  a  table  of  logarithms. 

The  logarithm  of  a  number,  is,  therefore,  the  exponent  of  the 
potoer,  to  lohich  it  is  necessary  to  raise  a  given  or  invariable 
number,  in  order  to  produce  the  proposed  number. 

Thus  in  the  equation  a'-==y,  x  is  the  logarithm  of  y ;  in  like 
manner  in  the  equation  2^  =  64,  6  is  the  logarithm  of  64.  The 
logarithm  of  a  number  is  indicated  by  writing  before  it  the  first 
three  letters  of  the  word  logarithm,  or  more  simply  by  placing 
before  it  the  letter  L. 

The  invariable  number,  from  which  the  others  are  formed  JB 
called  the  base  of  the  table.  It  may  be  taken  at  pleasure  either 
greater  or  less  than  unity,  but  should  remain  the  same  for  the 
formation  of  all  numbers,  that  belong  to  the  same  table. 

Since  a°=  1,  and  d^  =  a,  whatever  number  may  be  assumed 
for  the  base  of  the  table,  the  logarithm  of  the  base  will  be  unity 
and  the  logarithm  of  unity  will  be  0. 

198.  We  proceed  to  show  the  properties  of  logarithms  in  rela- 
tion to  numerical  calculations. 


240  ELEMENTS    OF   ALGEBRA. 

1.  Let  there  be  the  series  of  numbers  y,  y\  y'\  ....  to  be 
multiplied  together.  Let  a  represent  the  base  of  a  system  of 
logarithms,  which  we  suppose  already  calculated,  and  let  z,  a;', 
x"  .  .  be  the  logarithms  of  2/,  y\  y",  .  .  . ;  by  the  definition  of  a 
logarithm  we  have 

y  =  a%y'=a'\y"  =  a"'; 
multiplying    these    equations    member    by   member,   we    have 

yy'y"=a'+''+'", 
whence  log  yy'y"  =^X'\-  x'  -\-  x"  =  log  y  +  log  y'  •\-  log  y'\ 

That  is,  the  logarithm  of  a  'prodwct  is  equal  to  the  sum  of  the 
logarithms  of  the  factors  of  this  product. 

If  then  a  multiplication  be  proposed,  we  take  from  a  table  of 
logarithms  the  logarithms  of  the  numbers  to  be  multiplied ;  the 
sum  of  these  logarithms  will  be  the  logarithm  of  the  product 
sought.  Seeking  therefore  this  logarithm  in  the  table,  the  num- 
ber corresponding  to  it  will  be  the  product  sought.  Thus  by 
means  of  a  table  of  logarithms  addition  may  he  made  to  take  the 
place  of  multiplication, 

2.  Let  it  be  required  to  divide  the  number  y  by  the  number 
y' ;  let  a;,  x'  be  the  logarithms  of  these  numbers,  we  have  the 
equations 

dividing  these  equations  member  by  member,  we  have 

whence  log  ~,  =  x  —  x'  =■  log  y  —  log  y'. 

That  is,  the  logarithm  of  a  quotient  is  equal  to  the  difference 
between  the  logarithm  of  the  divisor  and  that  of  the  dividend. 

If  then  it  be  proposed  to  divide  one  number  by  another,  from 
the  logarithm  of  the  dividend  we  subtract  the  logarithm  of  the 
divisor,  the  result  will  be  the  logarithm  of  the  quotient ;  seeking 
therefore  this  logarithm  in  the  tables  the  number  corresponding 
will  be  the  quotient  sought.  Thus,  by  means  of  a  table  of  loga- 
rithms, subtraction  may  be  made  to  take  the  place  of  division. 


THEORY   OF   LOGARITHMS.  241 

3.  Let  it  next  be  required  to  raise  the  number  y  to  the  power 
denoted  by  tw,  we  have  the  equation         a*  =  y ; 

raising  both  members  to  the  mth  power,  we  have 

whence  the  logarithm  oi  y"^  =  mx  =■  m\og  y. 

That  is,  the  logarithm  of  any  power  of  a  number  is  equal  to  the 
product  of  the  logarithm  of  this  number  by  the  exponent  of  the 
power. 

To  form  any  power  whatever  of  a  number  by  means  of  a  table 
of  logarithms,  we  multiply,  therefore,  the  logarithm  of  the  pro- 
posed number  by  the  exponent  of  the  power,  to  which  it  is  to  be 
raised;  the  number  in  thfe  table  corresponding  to  this  product, 
will  be  the  power  sought. 

4.  Again,  let  it  be  required  to  find  the  n\k  root  of  y.  We 
have  as  before  a*  =  y ; 

whence  taking  the  wth  root  of  both  members,  we  have 

a"  =  y"  ;  whence  log  2/"  =  -  =  — 2_^ 

That  is,  the  logarithm  of  the  root  of  any  degree  whatever  of  a 
number  is  equal  to  the  logarithm  of  this  number  divided  by  the 
index  of  the  root. 

Thus  by  the  aid  of  a  table  of  logarithms  a  number  Tnay  he 
raised  to  a  power  by  a  simple  multiplication,  and  its  root  Tnay  he 
extracted  by  a  simple  division. 

FORMATION   OF   TABLES. 

199.  The  properties  of  logarithms  demonstrated  above  are 
altogether  independent  of  the  number  a  or  their  base.  We 
may  therefore  form  an  infinite  variety  of  tables  of  logarithms 
by  putting  for  a  all  possible  numbers  except  unity. 

If  it  be  required  to  construct  a  table  of  logarithms  the  base  of 
which  is  2,  in  the  equation  2'  =  y,  we  make  y  equal  successively 
to  the  numbers  1,  2,  3  ...  .,  and  determine  by  the  methods 
explained,  art.  195,  the  values  of  x  corresponding. 

We  thus  obtain  the  values  of  x  exactly,  if  y  be  a  perfect  power 
of  2,  or  otherwise  with  such  degree  of  approximation  as  we  please 


242 


ELEMENTS    OF   ALGEBRA. 


To  calculate  the  logarithm  of  3,  for  example,  we  have  the 

equation  2*  =  3,  from  which  we  deduce 
1 


x=l 


l-f 


1 


1 


Whence  stopping  at  the  fourth  integrant  fraction,  and  forming 

the  reduction  corresponding,  we  have  x=  j^,  or  reducing  this 

last  to  a  decimal  we  have  z=  1.583  accurate  to  the  third  deci- 
mal figure, 

200.  In  the  calculation  of  a  table  of  logarithms,  it  will  he 
sufficient  to  calculate  directly  the  logarithms  of  the  prime  num- 
bers 1, 2,  3,  5  . . . ,  the  logarithms  of  compound  numbers  may  then 
be  obtained  by  adding  the  logarithms  of  the  prime  factors,  which 
enter  into  them.  To  find  the  logarithm  of  35,  for  example,  we 
have  35  =  5  X  7 ;  whence  log  35  =  log  5  -f-  log  7 ;  having  al- 
ready calculated  the  logarithms  of  5  and  7,  the  logarithm  of  35 
will  be  found  therefore  by  adding  the  logarithm  of  5  to  that  of  7. 

Since  moreover  the  logarithm  of  a  fraction  will  be  equal  to  the 
logarithm  of  the  numerator  minus  the  logarithm  of  the  denomi- 
nator, it  will  be  sufficient  to  place  in  the  tables  the  logarithms  of 
entire  numbers. 

201.  Below  we  have  a  table  of  logarithms  of  numbers  from 
1  to  30  inclusive,  the  base  of  the  system  is  2,  and  the  logarithms 
are  calculated  to  4  places  of  decimals. 


N. 

Log. 

N. 

Log.  |N. 

Log. 

1 

0.0000 

11 

3.4594 

21 

4.3922 

2 

1.0000 

12 

3.5849 

22 

4.4594 

3 

1.5849 

13 

3.7000 

23 

4.5235 

4 

2.0000 

14 

3.8073 

24 

4.5849 

5 

2.3219 

15 

3.9065 

25 

4.6438 

6 

2.5849 

16 

4.0000 

26 

4.7000 

7 

2.8*073 

17 

4.0874 

27 

4.7548 

8 

3.0000 

18 

4.1699 

28 

4.8073 

9 

3.1699 

19 

4.2479 

29 

4.8577 

10 

3.3219 

20 

4.3219 

30 

4.9065 

THEORY   OF   LOGARITHMS.  243 

202.  The  most  convenient  number  for  a  base  to  a  system  of 
logarithms,  and  the  one  employed  in  the  construction  of  the 
tables  in  common  use  is  10. 

If  in  the  equation  10*  =  y  we  make  successively 


a:=:0, 

1,        2,        3,            4  . 

we  have 

y=l, 

10,     100,  1000,     10000  . 

Again  if 

we  make 

x  =  0, 

-1,  -2,    -3,    -4  .  . 

we  have 

y=i, 

1111 
10'     100'    1000'  10000' 

Therefore  in  a  table  of  logarithms,  the  base  of  which  is  10, 
1**.  the  logarithms  of  numbers  greater  than  unity  are  positive 
and  go  on  increasing  from  0  to  infinity.  2°.  The  logarithms  of 
numbers  less  than  unity  are  negative,  and  their  absolute  values 
are  so  much  the  greater  as  the  fractions  are  smaller ;  whence  if 
we  take  a  fraction  less  than  any  assignable  quantity,  the  loga- 
rithm of  this  fraction  will  be  negative,  and  its  absolute  value  will 
be  greater  than  any  assignable  quantity.  On  this  account  we 
say  that  the  logarithm  of  0  is  an  infinite  negative  quantity. 
3°.  The  logarithms  of  all  numbers  below  10  are  fractions;  the 
logarithms  of  numbers  between  10  and  100  are  1  and  a  fraction ; 
the  logarithms  of  numbers  between  100  and  1000  are  2  and  a 
fraction ;  those  of  numbers  between  1000  and  10000  are  3  and  a 
fraction ;  and  in  general,  the  whole  number  which  precedes  the 
fraction  in  the  logarithm  is  less  by  one  than  the  number  of 
figures  in  the  number  corresponding  to  the  logarithm.  On  this 
account  it  is  called  the  index  or  characteristic  of  the  logarithm, 
since  it  serves  to  indicate  the  order  of  units,  to  which  the  number 
corresponding  to  the  logarithm  belongs;  Thus  in  the  logarithm 
3.75527  the  characteristic  3  shows  that  the  number  correspond- 
ing to  this  logarithm  consists  of  4  figures  or  is  comprised  be- 
tween 1000  and  10000. 

203.  The  logarithm  of  a  number  being  given,  the  logarithm 
of  a  number  10,  100,  .  .  .  times  greater  is  found  by  adding 
1,  2,  .  .  .  units  to  the  characteristic  only ;  indeed  log 
{y  X  10")  =  log  2/  +  log  10"  =  log 2/  +  7i; 


24^  ELEMENTS    OF   ALGEBRA. 

whence  it  will  be  sufficient  to  add  n  units  to  the  logarithm  of  y  in 
order  to  obtain  the  logarithm  of  a  number  10"  times  as  great ;  an 
addition  which  may  be  performed  upon  tbe  characteristic  only. 

Conversely,   log :^  =  logy  —  log  10"  =  log 2/  —  n;  thus  it  is 

sufficient  to  subtract  n  units  from  the  logarithm  of  y^  in  order  to 
find  the  logarithm  of  a  number  10"  times  smaller  than  y. 

204.  The  fractional  parts  of  logarithms  in  the  tables  are  ex- 
pressed by  decimals.  From  what  has  been  said  the  decimal  part 
of  the  logarithm  of  a  number  will  be  the  same  for  this  number 
multiplied  or  divided  by  10,  100,  ...  On  this  account  the  sys- 
tem of  logarithms,  the  base  of  which  is  10,  is  more  convenient 
than  any  other  system,  since  we  have  frequent  occasion  to  multi- 
ply or  divide  by  10,  100,  .  .  .  operations  reduced  in  this  case  to 
tfie  simple  addition  or  subtraction  of  units. 

205.  Since  the  characteristic  of  the  logarithm  may  be  easily 
determined  by  the  number,  and  the  number  of  figures  in  the 
number  by  the  characteristic  of  the  logarithm,  it  is  usual  to 
omit  the  characteristic  in  the  tables  to  save  the  room.  It  is 
also  convenient  to  omit  it ;  because  the  same  decimal  part  with 
different  characteristics  forms  the  logarithms  of  several  different 
numbers. 

206.  Having  already  calculated  a  system  of  logarithms,  it 
will  be  easy  from  this  to  form  as  many  other  systems  as  we 
please. 

Indeed,  let  N  designate  any  number  whatever,  log  N  its  loga* 
rithm  in  the  system  the  base  of  which  is  a,  X  its  logarithm  in  a 
different  system  the  base  of  which  is  3,  we  have 

i^  =  N. 
Taking  the  logarithms  of  both  members  of  this  equation  in  the 
system,  the  base  of  which  is  c,  we  have 
X .  log  *  =  log  N  J 

whence  X  =  ,^  , . 


TflEORY    OF   LOGARITHMS.  245 

Having  calculated  therefore  a  set  of  tables  for  a  particular 
base,  to  find  the  logarithm  of  a  number  in  a  proposed  system 
with  a  different  base,  we  take  from  the  tables  already  calculated 
the  logarithm  of  the  number ,  and  also  the  logarithm  of  the  base 
of  the  proposed  system;  the  former  of  these  logarithms,  divided  by 
the  latter,  will  give  the  logarithm  of  the  number  in.  the  proposed 
system. 

The  logarithm  of  6,  for  example,  in  the  system,  the  base  of 
which  is  10,  is  .77815,  and  that  of  3  is  .47712 ;  the  logarithm 
of  6,  therefore,  in  the  system  the  base  of  which  is  3,  will  be 

^=1.63093. 
47712 

loffN 

207,  The  expression  X  =  may  be  put  under  the  form 

X  =  I — r  log  N.     Thus  having  already  formed  a  table  of  loga- 
logd 

rithms,  the  base  of  which  is  a,  to  construct  from  this  a  new  table, 

the  base  of  which  shall  be  ^,  we  multiply  the  logarithms  of  the 

first  table  by  the  quantity  -, — r.     This  quantity  by  means  of 

which  we  are  enabled  to  pass  from  the  old  to  the  new  table,  is 
called  the  modulus  of  the  new  table  in  relation  to  the  old. 

MODE    OF    USING   THE    TABLES. 

208.  As  it  is  impossible  to  place  in  the  tables  the  logarithms 
of  all  numbers,  it  is  usual  to  place  in  them  the  logarithms  of 
numbers  from  unity  to  within  a  certain  limit  In  what  follows  it 
is  supposed,  that  the  student  has  in  his  hands  tables  containing 
the  logarithms  of  entire  numbers  from  1  to  10000. 

In  order  to  use  such  a  set  of  tables,  we  have  the  two  following 
questions  to  resolve,  viz.  1°.  Any  number  whatever  being  given, 
to,  find  its  logarithm.  2**.  Any  logarithm  being  given,  to  find 
the  number  tohich  corresponds  to  it. 

The  following  examples  will  exhibit  the  method  of  resolving 
these  questions. 

1.     Let  it  now  be  proposed  to  find  the  logarithm  of  9748 


246  ELEMENTS    OF   ALGEBRA. 

Seeking  the  proposed  in  the  column  of  numbers,  against  it  in 
the  column  of  logarithms  we  find  98892 ;  this  will  be  the  deci- 
mal part  of  the  logarithm ;  or,  as  is  the  case  with  most  tables, 
if  the  column  of  numbers  contain  but  three  places  of  figures, 
we  look  for  974,  the  first  three  figures  of  the  proposed,  in  the 
first  column,  and  at  the  top  of  the  table  we  look  for  the  fourth 
figure  8 ;  directly  under  the  8  and  in  the  same  line  with  974, 
we  find  the  decimal  part  98892  as  before ;  then  since  the  pro- 
posed consists  of  four  places,  the  characteristic  will  be  3,  thus 
log  9748  =  3.98892. 

2.  Let  it  be  required  to  find  the  logarithm  of  76.93.  Re- 
moving for  the  moment  the  decimal  point,  we  find  as  above 
log  7693  =  3.88610,  whence,  art.  203,  subtracting  2  units  from 
the  characteristic  3  of  this  logarithm,  we  shall  have  the  logarithm 
of  the  proposed ;  thus  log  76.93  =  1.88610. 

3.  To  find  the  logarithm  of  .75.  The  logarithm  of  this  num- 
ber may  be  presented  under  two  different  forms.     Writing  it 

75 

in  the  form  of  a  vulgar  fraction,  it  becomes  — rr-.  The  loga- 
rithm of  75  is  1.87506,  and  that  of  100  is  2.00000;  whence 
subtracting  the  logarithm  of  the  denominator  from  that  of  the 
numerator,  art.  198,  we  have  —  12494  =  log  .75.  This  loga- 
rithm, being   altogether  negative,  is   inconvenient  in  practice; 

it  will  be  observed,   however,   that  .75  =  -rTrTrX75;   whence 

log  .75  =  log  ^  +  log  75  =  -  2  +  1.87506, 

=  _  1-^87506, 
or  placing  the  sign  —  over  the  1  to  show  that  the  characteristic 
only  is  negative,  we  have  log  .75  ==  1.87506. 

This  last  form  of  the  logarithm  of  the  proposed  is  derived,  it 
will  be  perceived,  immediately  from  the  continuation  of  the  prin- 
ciple, art.  203,  according  to  which  the  logarithm  of  a  number 
10,  100  .  .  .  times  less  than  a  proposed  number  is  found  by  sub- 
tracting 1,  2  .  .  .  units  from  the  characteristic  of  its  logarithm. 


THEORY    OF   LOGARITHMS.  247 

Thus  since  the  logarithm  of  750  =  2.87506,  we  have 

log  75  =  1.87506 

log  7.5  =  0.87506 

log  .75  =1.87506 

log  .075  =  2.87506 

log  .0075  =  3.87506 
4 

4.  To  find  the   logarithm  of  - ;   we  have  log  4  =  .60206, 

og  5  =  .69897;   whence  subtracting  this  last  logarithm  from 

4 
the  former,  we  have  log  -  =  —  .09691,  in  which  the  logarithm 
o 
4 
is  entirely  negative.     But  -  reduced  to  a  decimal  becomes  .8, 

the  logarithm  of  which  is  1.90309,  the  characteristic  only  being 
negative. 

493 

5.  To  find  the  logarithm  of  54|- ;  we  have  5^1  =  -3- ;  log 

493  =  2.69285,  log  9  =  0.95424 ;  whence  subtracting  the  latter 

493 
logarithm  from  the  former,  we  have  log  —^  or  54^  =  1.73861. 

6.  To  find  the  logarithm  of  675437.  This  number  exceeds 
the  limits  of  the  table ;  its  logarithm,  however,  may  be  readily 
found.  The  greatest  number  of  places  in  a  number,  the  loga- 
rithm of  which  can  be  found  in  the  tables,  is  4 ;  separating  there- 
fore the  four  left  hand  figures  of  the  proposed  from  the  rest  by  a 
point,  we  consider  for  the  moment  those  on  the  right  as  deci- 
mals. The  logarithm  of  6754.37  is  comprised  between  the  loga- 
rithm of  6754  and  that  of  6755;  the  diflference  between  these 

37 

two  logarithms  is  .00007 ;  -jrr  of  this  difference  therefore  added 

to  the  less  logarithm  will  give  the  logarithm  of  6754.37  nearly ; 
thus  log  6754.37  =  3.82959 ;  whence  adding  2  units  to  the 
characteristic  of  this  last  to  obtain  the  logarithm  of  the  proposed, 
we  have  log  675437  =  5.82959. 

209.  We  proceed  next  to  the  second  of  the  proposed  questions, 
viz.  A  logarithm  being  given,  to  find  the  number  which  corres- 
ponds  to  it. 

1.  To  find  the  number  corresponding  to  the  logarithm  2.10449. 
The  decimal  part  of  this  logarithm  is  contained  in  the  tables ;  in 
the  left  hand  column  and  on  the  same  line  with  it  according  to 
the  arrangement  of  the  tables,  in  which  there  are  but  three  places 


His  ELEMENTS   OF   ALGEBRA. 

of  figures  in  the  column  of  numbers,  we  find  127,  and  at  the  top 
of  the  table  directly  over  it  we  find  2 ;  the  characteristic  of  the 
logarithm  being  2,  we  have  therefore  127.2  for  the  number  cor- 
responding to  the  proposed. 

2.  To  find  the  number  corresponding  to  the  logarithm  3.42674. 
This  logarithm  is  not  found  in  the  tables ;  it  is  comprised  however 
between  3.42667  the  logarithm  of  2671,  and  3.42684  that  oi 
2672 ;  the  diflference  between  these  two  logarithms  is  .00017,  the 
difference  between  the  proposed  and  3.42667  is  .00007 ;  we  have 
then  the  following  proportion  : 

.00017  :  1  :  :  .00007  :.41  nearly. 
The  number  corresponding  to  3.42674  is,  therefore,  2671.41. 

3.  To  find  the  number  corresponding  to  the  logarithm 
—  2.45379.  The  number  corresponding  to  this  logarithm  will 
be  comprised,  it  is  evident,  between  .01  and  .001 ;  to  obtain  this 
number  let  us  add  to  —  2.45379  a  sufficient  number  of  units  to 
make  it  positive,  5  for  example,  we  have  5  —  2.45379  =  2.54621 ; 
the  number  corresponding  to  this  last  is  351.73 ;  but  by  adding  5 
units  to  the  proposed  logarithm,  we  have  multiplied  the  number,  to 
which  it  belongs,  by  100000,  whence,  dividing  351.73  by  100000, 
we  have  .0035173,  the  number  corresponding  to  the  proposed. 

4.  To  find  the  number  corresponding  to  the  logarithm  3.86249. 
Adding  three  units  to  the  characteristic,  the  proposed  becomes 
0.86249,  the  number  corresponding  to  which  is  7.286 ;  whence,  as 
it  is  easy  to  see,  the  number  corresponding  to  3.86249  is  .007286 


SECTION   XXIV.— Application   of   the  Theory   of 
Logarithms. 

multiplication  and  division. 

1.  Let  it  be  required  to  multiply  872  by  .097. 

log  872  =  2.94052 

lo<T  .097  =  2.98677 


log  84.584  Ans.  1.92729 


APPLICATION   OF   LOGARITHMS.  249 

2.  Let  it  be  required  to  multiply  .857  by  .0093. 

log  .857  =  1.93298 
log  .0093  =  3.96848 

log  .00797  Am.  3.90146 

3.  Let  it  be  required  to  divide  5672  by  .0037. 

log  5672  =  3.75374 
log  .0037  =  3.56820 

log  1533000  Ans.  6.18554 

4.  Let  it  be  required  to  divide  .053  by  797. 

log  .053  =  2.72428  =  3  +  1.72428 
log  797=  2.90146 

log  .0000665  Ans.  5.82282 

To  render  the  subtraction  required  in  this  example  possible,  we 
change  the  characteristic  2  into  3  +  1,  which  has  the  same  value; 
this  furnishes  a  ten  to  be  joined  with  7  for  the  subtraction  of  9, 
the  left  hand  figure  of  the  decimal  part.  A  similar  preparation, 
it  is  evident,  must  be  made  in  all  cases  of  the  same  kind. 

FORMATION   OF   POWERS   AND   EXTRACTION   OF   ROOTS. 

210.  Let  it  be  required  to  find  the  5th  power  of  .125. 
log  .125  =  1.09691 
5 


log  .000030519  Ans.  nearly        5.48455 
2.  To  find  the  7th  power  of  .73. 

log*  .73  =1.86332 
7 


log  .11047  Ans.  nearly        7  +  6.04324=1.04324. 
3.  To  find  the  third  root  of  .01356. 
The  logarithm  of  .01356  is  2.13226.     The  negative  charac- 
teristic 2  of  this  logarithm  is  not  divisible  by  3,  the  index  of  the 
root  required,  neither  can  it  be  joined  to  the  positive  part  on 
account  of  the  different  sign.     If  however  we  add  —  1  +  1  to 


250  ELEMENTS    OF    ALGEBRA. 

the  characteristic,  which  will  not  alter  its  value,  it  becomes 
3  -|-  1 ;  the  negative  part  is  then  divisible  by  3,  and  the  1  being 
positive  may  be  joined  to  the  fractional  part,  we  have  then 

log  .01356  =  2.13226  =  3  +  1.13226; 
whence  dividing  by  3,  we  have 

1.37742  =  log  .23846  Ans.  nearly. 
In  all  cases,  if  the  negative  characteristic  is  not  divisible  by 
the  index  of  the  root  required,  it  must  be  made  so  in  a  similar 
manner. 

ARITHMETICAL   COMPLEMENT. 

211.  The  arithmetical  complement  of  a  logarithm  is  the  dif- 
ference between  this  logarithm  and  10;  thus  the  arithmetical 
complement  of  3.472584  is  10  —  3.472584  =  6.527416.  The 
arithmetical  complement  of  a  logarithm  is  obtained  by  subtract- 
ing the  right  hand  figure^  if  it  be  significant,  from  10,  and  the 
others  from  9. 

Let  it  be  proposed  to  find  the  value  of  z  in  the  expression 
x  =  l  —  l'J^l"  —  V"—V"' 
Z,  V,  I"  .  .  being  logarithms ;  this  expression,  it  is  evident,  may 
be  put  under  the  form 

:c  =  Z  +  (10  —  Z')  +  r  +  (10  —  V")  +  (10  —  I"")  —  30 ; 
that  is,  to  find  the  value  of  x,  we  take  the  sum  of  the  logarithms 
to  be  added  and  the  complements  of  the  logarithms  to  be  sub' 
tracted,  from  this  sum  subtract  as  many  times  10,  as  there  are 
complements  employed. 

'-^  Thus  when  there  are  several  multiplications  and  divisions  to 
be  performed  together,  by  using  the  complements  of  the  loga- 
rithms of  the  divisors  the  whole  may  be  reduced  to  the  addition 
of  logarithms. 

EXAMPLES. 


1.  To  find  the  value  of  x  in  the  expression 

■75  X  73  X  .056\t 
.7498  X  125.13  / 


_/37 


APPLICATION   OF    LOGARITHMS.  251 

log  3.75  0.57403 

log  73  1.86332 

log  .056  2.74819 

log  1.7498  Comp.  9.75701 

log  125.13  Comp.  7.90264 


18.84519 
Subtracting  next  20  from  the  characteristic,  and  taking  f  of 
the  remainder,  we  have  2.07532  =  log  .011803  Ans.      . 
2.  To  find  the  value  of  x  in  the  expression 

^_/132x(7.356)Y        Ans.  144.5972. 
^       (3.25)2-       / 

PROPORTIONS. 

212.  Let  it  be  required  to  find  the  fourth  term  of  the  proportion, 
of  which  the  numbers  963,  1279,  8.7,  are  the  first  three  terms. 

log  1279  3.10687 

log  8.7  0.93952 

log  963  Comp.  7.01637 

log  11.555  Am.  nearly      1.06276 

From  the  proportion  a:b::  c:d,  we  have  -  =  -  ; 

whence  log  a  —  log  ^  =  log  c  —  log  dj 

therefore  log  a  .  log  ^  :  log  c  .  log  d, 

that  is,  if  four  numbers  form  a  proportion^  their  logarithTns  vnU 

form  an  equidifference. 

EXPONENTIAL   EQUATIONS. 

213.  We  have  already  explained  a  method  for  finding  the 
value  of  X  in  the  equation  a'  =  b,  from  which  the  theory  of 
logarithms  is  derived ;  but  a  table  of  logarithms  being  once  con- 
structed, there  is  nothing  to  prevent  its  use  in  the  solution  of 
equations  of  this  kind. 

Let  it  be  required  to  find  the  value  of  x  in  the  equation 
3'=  15. 

Taking  the  logarithms  of  both  sides,  we  have 
a;  log  3  =  log  15; 


252  Elements  of  algebra. 

whence  ^._log  15  _  1-17609  _ 

Whence  a—  j^^^_    47712-"    ^^ 

The  division  required  in  this  example  may  be  performed,  it  is 
easy  to  see,  by  subtracting  the  logarithm  of  .47712  from  that  of 
1.17609,  as  in  the  case  of  any  other  numbers. 

PROGRESSION    BY    QUOTIENT. 

214.  Logarithms  are  particularly  useful  in  the  solution  of 
questions  in  progression  by  quotient. 

Let  it  be  proposed  to  find  the  20th  term  in  the  progression 
3    9    27 
•  2  '•  4  '  8  *  • 
Putting  u  for  the  last  term  of  a  progression  by  quotient,  we 
have,  art.  176,  ** 

u  =  aq"~^;  whence,  log  u  =  log  a-\-  {n  —  1)  log  q. 
We  have,  therefore,  for  the  20th  term  in  the  progression  pro- 
posed 

log  t^  =  log  1  +  19  (log  3  —  log  2)  =  19  (log  3  —  log  2) 
the  term  required  will  therefore  be  2216.84  to  within  .01. 

Let  it  be  required  next  to  insert  between  the  numbers  2  and 
15  fifty  mean  proportionals ;  We  have  for  the  ratio,  art.  184, 


-          ,             losri  —  logfl 
whence  locf  q  =  -^ ,    , 


in  the  question  proposed,  we  have,  therefore, 
,  W  15  —  loff  2 

'°gg=  51  ■ 

or,  performing  the  calculations,  we  obtain 
^=1.040286. 
215.  Let  it  be  required  to  find  the  sum  of  the  first  ten  terms  in 
the  progression  -rf  5  .  15  .  45  .  .  . ;  we  have,  art.  178, 

8  =  —^^ — :; — ;  whence 
^—1 

log  S  =  log  a  +  log  (g"  —  1)  —  log  {q  —  1). 
Applying  this  formula  to  the  proposed  question,  we  have 

log  S  =  log  5  +  log  (3^°  —  1)  —  log  (3  —  1). 
Calculating  3^°  by  logarithms,  we  have 
log  3^°  =  10  X  loe-  3, 


A/'^'i.iUATlON   OF   LOGABITHMS.  253 

from  which  we  obtain  3'°  =  59048 ; 

whence  log  S  =  log  5  +  log  (59048  —  1)  —  log  2, 

or,  performing  the  calculations,  we  obtain  147620  for  the  sum 

required. 

Let  it  be  proposed  next  to  find  the^  number  of  terms  in  the 
progression  of  which  the  first  term  is  3,  the  ratio  2,  and  the  last 
term  6144. 

From  the  formula  w  =  ag^""^  we  have 

log  u  =  log  a-{-{n  —  1)  log  q  ; 

,  ,    ,   \q^u  —  Wa 

whence  w  =  1  H — 2_- 2-_, 

\ogq 

Applying  this  formula  to  the  proposed  question,  we  have 

log  2 

216.  Let  us  take  next  the  progression 

rrr  d'-^'^cd'.  e:f:  g  .  . 
from  the  nature  of  the  progression,  we  have 
a       b       c       d       f 
b^^  c       d       e       g* 

whence        log  -  =  log  -  =  log  ^  =  log  -  .  .  .  . 
b  c.  a  e 

wherefore,   log  a  —  log  i  =  log  b  —  log  c  =  log c  —  \ogd=  ,  . 
from  this  last  we  have 

-7-  log  a  .  log  i  .  log  c  .  log  d  .  .  .  , 
If,  therefore,  the  numbers  a^  b,  c,  d  ,  .  form  a  progression  by 
quotient,  their  logarithms  will  form  a  progression  by  difference. 
Logarithms  may  therefore  be  defined  a  series  of  numbers  in 
arithmetical  progression  corresponding  term  to  term  to  aTwther 
series  of  numbers  in  geometrical  progression.  This  is  the  defini- 
tion of  logarithms  given  in  arithmetic. 

COMPOUND  INTEREST. 

217.  One  of  the  most  important  applications  of  logarithms  ia 
to  questions  upon  the  interest  of  money. 

Interest  is  of  two  kinds,  simple  and  compound.  If  interest  be 
paid  upon  the  principal  only,  it  is  called  simple  interest;  but  if 


254  ELEMENTS   OF   ALGEBRA. 

the  interest,  as  it  becomes  due,  be  added  to  the  principal,  and  in 
terest  be  paid  upon  the  whole,  it  is  then  called  compound  interest. 

We  have  already  investigated  formulas  for  simple  interest. 
Let  it  now  be  proposed  to  determine  what  sum  a  given  principal 
p  will  amount  to,  in  a  number  n  of  years,  at  a  given  rate  r  at 
compound  interest. 

The  amount  of  unity  for  one  year  will  be  1  -[-  r ;  that  of  p 
units  will  be  therefore  p{\-\-r). 

For  the  second  year  p{^-\-r)  will  be  the  principal,  and  its 
amount  will  he  p{\-\-  r)  (1  -\-r)  oi  p{l-\-  rf. 

The  original  sum  p  therefore  at  the  end  of  the  second  year 
will  amount  to  p{l-\-rY'  In  like  manner  at  the  end  of  the 
third  year  it  will  amount  to  p(l  +r)^;  whence  putting  A  for 
the  amount  required,  we  have 

A=p{l  +  r)\ 
This  is  a  general  formula  for  compound  interest;  taking  the 
logarithms  of  both  sides  we  have, 

log  A  =  log^  -f"  71  log  (1  -f-  r). 

Let  it  be  proposed  to  determine  what  sum  $30000  will  amount 
to,  in  30  years,  at  5  per  cent,  compound  interest. 

We  have        log  A  =  log  30000  +  30  log  1 .05, 
whence,  we  obtain  $  1^9658.27,  Ans. 

218.  The  equation  A=p{l-\-rY  contains  four  quantities 
A,  p,  r,  and  n^  any  one  of  which  may  be  determined,  when  the 
others  are  known.  It  gives  rise  therefore  to  the  four  following 
questions,  viz. 

1**.  To  determine  A,  when  p,  r,  and  n  are  given,  or  the  princi' 
palf  rate,  and  number  of  years  being  given,  to  find  the  amount. 

This  question  we  have  already  solved. 

2°.  To  determine  p  lohen  A,  r,  and  n  are  given,  or  to  find 
what  principal  put  at  compound  interest  will  amount  to  a  given 
sum,  in  a  certain  number  of  years,  at  a  given  rate. 

Resolving  the  general  equation  with  reference  to  p,  we  have 
_       A 

or  by  logarithms        log  j9  =  log  A  —  w  log  ( 1  +  ^)' 


APPLICATION   OF    LOGARITHMS.  255 

3".  To  determine  r,  when  A,  p,  and  n  are  hnown^  that  is,  to 
find  at  what  rate  a  given  sum  must  be  put  at  compound  interest^ 
in  order  to  amount  to  another  given  sum  in  a  given  time. 

Resolving  the  general  equation  with  reference  to  r,  we  have 

or  by  logarithms        log  (1  -f-  r)  =  — ^ —. 

Having  by  means  of  this  last  determined  the  value  of  1  -|-  r, 
that  of  r  will  be  easily  found. 

4°.  To  determine  n,  when  A,  p,  and  r  are  given,  that  is,  to 
find  for  what  time  a  given  sum  must  he  put  at  compound  interest 
at  a  certain  rate  in  order  to  amount  to  a  given  sum. 

Making  n  the  unknown  quantity  in  the  general  formula,  we 

obtain 

log  A  —  log^ 

"""     log(l  +  r)'      , 
If  it  be  asked  what  must  be  the  value  of  n  in  order  that  the 
sum  at  interest  may  be  doubled,  tripled,  &c. ;  we  put  in  the 
general  formula  A.  =  kp,  k  denoting  1,  2,  3  .  .  .  ,  we  thus  have 

Tcp=p{l  +  rY;  whence  71  =  j^^^i-p^; 

n  is  therefore  independent  of  p,  that  is,  whatever  the  sum  put 
out,  it  will  be  doubled,  tripled,  &c.  in  the  same  time. 

EXAMPLES. 

1.  What  the  amount  of  $1000  for  25  years  at  5  per  cent, 
compound  interest  ?  Ans.  $3386. 

2.  "What  will  $600  amount  to  in  6  years  at  4J  per  cent, 
compound  interest,  supposing  the  interest  to  be  payable  half 
yearly?  Ans.  $783.63. 

3.  In  a  certain  province  there  are  at  present  200000  inhab- 
itants. If  the  population  increases  -^jj  part  yearly,  what  will  it 
be  100  years  hence  ?  Ans.  1448927,  nearly. 

4.  How  much  money  must  be  placed  out  at  compound  inter- 
est to  amount  to  $1000  in  20  years,  the  interest  being  5  per 
cent.?  Ans.  $376.89. 


256  ELEMENTS    OF   ALGEBRA. 

5.  A  sum  of  $201.22  is  payable  12  years  hence  without  inter- 
est. What  sum  put  out  at  6  per  cent,  compound  interest  will  be 
sufficient  to  meet  the  payment  at  the  end  of  that  time  ? 

Ans.  $100.00. 

6.  The  sum  of  $  500  put  out  at  5  per  cent,  compound  inter- 
est has  amounted  already  to  $900.  How  long  has  it  been  at 
interest  ?  Ans.  12.04  years. 

7.  A  capital  of  $3200  having  been  at  compound  interest  for 
80  years  has  amounted  to  $34050.84,  at  what  rate  per  cent,  was 
it  put  out  ?  Ans.  3  per  cent. 

8.  In  what  time  will  a  principal  be  doubled  at  5  per  cent.? 
In  what  time  will  it  be  tripled  at  6  per  cent.  ? 

ANNUITIES. 

219.  An  annuity  is  a  sum  of  money  payable  yearly  for  a  cer 
tain  number  of  years  or  forever. 

Let  it  be  proposed  to  determine  what  sum  must  be  put  at 
interest  to  pay  an  annuity  of  b  dollars  for  n  years,  the  interest 
being  reckoned  at  the  rate  r  compound  interest. 

According  to  the  rule  for  compound  interest,  the  amount  of 
the  first  payment,  at  the  expiration  of  the  n  years,  will  be 
i(l  +  r)'*~\    the    amount    of   the    second    payment    will    be 

^(1  _|_  rY~\  that  of  the  third  will  be  ^(1  +r)"-' the 

last  payment  will  be  b.     Putting  A  for  the  sum  placed  at  interest 
for  the  payment  of  the  annuity,  its  amount  at  the  end  of  the 
n  years  will  be  A(l-(-r)";  we  shall  have  therefore 
A{l  +  rY  =  b{l  +  rY-'  +  b{l  +  rr-^-\-b{l  +  rY-K  .  .  h, 
but  the  second  member  of  this  equation  forms,  it  is  evident,  a 

progression  by  quotient  the  ratio  of  which  is      .     ,  or,  the  ordei 

of  the  series  being  reversed,  1  +  r  ;  taking  its  sum  we  have 

A(l  +  r)-  =  ^[<^+;>"-^; 

whence      •  ^^m+rfU 

r  (1  +  r)* 


PRAXIS  257 

This  equation  gives  rise  also  to  four  different  questions,  ac- 
cording as  we  make  A,  b^  r,  or  n  the  unknown  quantity.  The 
following  examples  exhibit  particular  cases  of  these  questions. 

1.  A  man  wishes  to  purchase  an  annuity  which  shall  afford 
him  $  1500  a  year  for  12  years.  What  sum  must  he  deposit  in 
the  annuity  office  to  produce  this  sum,  supposing  he  can  be 
allowed  7J  per  cent,  interest?  Ans.  $11602.91. 

2.  A  man  purchased  an  annuity  for  15  years  for  $100000. 
How  much  can  he  draw  annually,  the  interest  being  reckoned  at 
5  per  cent.?  Ans.  $9634.22. 

3.  A  man  has  property  to  the  amount  of  $  34580,  which  yields 
him  an  income  of  4  per  cent.  His  annual  expenses  are  $2000. 
How  long  will  his  property  last  him  ?      Ans.  30  years  nearly. 


SECTION  XXV.— Praxis. 


I. — equations  of  the  first  degree. 

1  1 

1.     Given  {x  +  40)^  =  10  —  x^,  to  find  the  value  of  x. 


Squaring  both  sides  of  the  equation,  we  have 

a:_j.40=100  — 20a:*  +  a; 
whence  a:  =  9. 

2.  Given  {x  —  16)^  =  8  —  a;^,  to  find  the  value  of  x. 

Ans.  a;  =  25. 

.    ^.        a:* +  28       a:* +  38  ,    .    ,.         ,        - 

3.  CjTiven  —^ =  —r-^ ,  to  find  the  value  of  x. 

a;^  +  4         a;5  _j_  6 

Freeing  from  denominators  and  reducing,  we  have  16  =  8a;  , 
whence  a;  =  4. 

(9a:)*— 4       15  + (9a:)*       ^   ,  ^        , 

4.  Given  ^ — j =  — r-i— ^ ,  to  find  the  value  of  x. 

a;*  +  2  a:^+40 

Ans.  a:  =  4. 
11  4 

5.  Given  (2  +  a;)^  +  a:^  = j,  to  find  the  value  of  x. 

(2  + a:)* 
17  Ans.  a:=|. 


258  ELEMENTS   OF    ALGEBRA. 

6.  Given  z^  A-  {x  —  9)^  = j-,  to  find  the  value  of  sc* 

{x-9f 

Ans.  a;  =  25. 

n. — INCOMPLETE    EQUATIONS    OF   THE    SECOND   DEGREE. 

1.  Given  a:^  +  ?/^=  189)  ,    ^    ,,         ,  .         , 

1    «      ,  '   % 180  (  values  ol  x  and  y. 

Adding  3  times  the  second  equation  to  the  first  and  extracting 
the  third  root,  we  have     a;  -|-  2/  =  9, 

but       C(Py'{-X'i/'  =  xy{x-\-y),  whence  9^:2/ =180, 

and  xy  =  20;  subtracting  4  times  this  last  from  the  equation 
a:  -j-  2/  =  9  raised  to  the  square,  and  extracting  the  square  root 
of  both  sides  of  the  remainder,  we  obtain  x  —  yz=:l;  whence 
2:  =  5,  2/  =  4. 

2-  ^'21  ^'^ll^ll]  to  findthe  values  of  x  and  y. 

Ans.  a;  =  dz  2,  y  =  db  4. 
^^   1      "^   2ZI 18  [  to  find  the  values  of  a;  and  y. 

Ans,  a;  =  zfc  9,  3/  =  ^t:  3. 
\    o  ^         2 IZ  1 A  ( to  find  the  values  of  x  and  y. 

Ans.  a:  =  4  or  — 2,  y  =  2  or  — 4. 

13    1 

5.  Given  x^-\-y^  = 

/>  -^  V  to  find  the  values  of  x  and  y. 

anda;v  = 

a;  — 2/J 

Ans.  a;  =  3  or  — 2,  y  =  2  or  — 3. 

6.  Given  x   +y^  =  13)  ^^  ^^^  ^^^  ^^^^^^  ^^ ^  ^^^  ^^ 

and  x^  -^y^  =   5  ) 
Squaring  the  second  equation 

a;^  +  2a:*2/*+2/^  =  25 

but  ^ +y^  =  ]3 

whence  by  subtraction  2  a;  y    =  12 

subtracting  this  last  from  the  first  equation 

x^-.2x^y^+y^=l 
whence  a:   —  y^  =  ±  1 


PRAXIS. 


from  which  compared  with   the   second   equation,  we   obtain 
a:  =  27  or  8,  y  =  S  or  27. 

alTJ^f  =  34  I  *°  ^"^  ^^®  ^^^^^^  °^  ^  ^'^^  ^• 

Ans.  2;  =  5  or  3,  y  =  3  or  5. 

IndJy'  tf  ^  333  |  ^°  ^""^  ^®  ^^^^®'  °^  ^  ^"^  y* 

Ans.  a:  =  2,  y==3. 

9.  Given  a;^  + 2/^  =  20  K    i.   i  ,i,       i         r        j 

2         1  Mo  in^d  the  values  of  x  and  ^. 

anda;^  +  2/^=    6  ) 

Ans.  x  =  ±8  or  ±  V8,  2/  =  32  or  1024. 

10.  Given  x-{'X  y^  -\-y=    19  /  to  find  the  values 

and  ar^  +  a:2/-j-^=  133  j        of  a;  and  2/. 

Dividing  the  second  equation  by  the  first,  we  have 

x-'X^y^+y  =  7; 
adding  this   last   to   the   first   and    dividing   by  2,   we   obtain 
a:  -|-  2/  ==  13 ;  subtracting  it  from  the  first,  dividing  by  2  and 
squaring  both  sides  of  the  result,  we  have  xy  =  36;  comparing 
the  equations  thus  obtained,  we  have  a:  =  9  or  4,  y  =  4  or  9. 

11.  Given      ^^     ==48^ 


4 
x^y 

and  fi^  = 


i2f  =  24[ 
x^  J 


to  find  the  values  of  x  and  y. 


Ans.  a;  ^36,  y  =  4. 

12.  Given  a:^  —  v*  =  369)  ,    n  a  ,u       ^         c        a 

lo ^ 9(^®  ^^  *"®  vames  of  x  and  y. 

Ans.  x  =  :^5i  y  =z=  ^  4. 

and^a.-^Z-f  J^=  12  |  ^^  ^"^  ^^®  ^^^^^^  °^ ^  ^"^  ^• 
Ans.  a;  =  2  or  1,  y  =  1  ox  2. 

III.— COMPLETE   EQUATIONS   OF   THE    SECOND   DEGREE. 

1.  Given  a;   -|-  ^  =  ''^^j  ^^  find  the  values  of  a;. 

n         1  r       .1.  f    f      f   I    1        3025 

Completing  the  square,  a:*4"^+T  =  —j- 1 

extracting  the  root      ^  a;^  -|-  -  =  ±  -q-  ; 

8 

from  which  we  obtain  z  =  243  or  ( —  28)  . 


260  ELEMENTS   OF   ALGEBRA. 

2.  Given  3^  —  a;=  56,  to  find  the  values  of  ar. 

Ans.  a:  =  4,  or  (— 7f. 

3.  Given  3  a;*  +  a;^  =  3104,  to  find  the  values  of  x. 

Ans.  a;  =  64,  or  (— ^7-)*. 

4.  Given  z^  -{-z^  =  Qfx^,\.o  find  the  values  of  x. 

Ans.  a:  =  2,  or  — 3. 

5.  Given  ar  —  a;  =  2  a:  ,  to  find  the  values  of  x, 

Ans.  ar  =  4  or  1. 

6.  Given  a;  -|-  5  —  (a;  -}-  5)^  =  6,  to  find  the  values  of  x. 
Completing  the  square,  iC  +  5~(a:-{-5)^+7==X' 

extracting  the  root  {x-\-5)^  —  -  =  2t:oJ 

from  which  we  obtain  a:  =  4,  or  —  1. 

7.  Given  (x  +  12)  ^  +  (a;  +  12)*  ==  6,  to  find  the  values  of  x, 

Ans.  x  =  ^y  or  69. 

8.  Given  a:  +  16  —  7  (a:  +  16)*  =  10  —  4  (a;  +  16)^  to  find 
the  values  of  x,  Ans.  a;  =  9,  or  —  12. 

9.  Given  a;^  +  {5x  +  a;*)^  =  42  —  5a:,  to  find  the  values  of  a:. 

Ans.  a:  =  4,  or  —  9. 

10.  Given  a:^  — 2a;  +  6(ar^  — 2a:  +  5)*  =  11,   to   find   the 
values  of  x. 

Adding  5  to  each  member 

ar'  — 2a: +  5  + 6  (a:*  — 2a; +  5)*  =  16, 
completing  the  square 

ar'  — 2a:  +  5  +  6(ar'  — 2ar  +  5)*  +  9  =  25, 
extracting  the  root  and  reducing,  we  obtain 
a:=l,  or=fc2yv/15. 

11.  Given  9a;  — 4ar»+ (4ar'  — 9a:  + 11)*  =  5,  to  find  the 
Talues  of  X,  Ans.  a;  =  2,  or  \, 

12.  Given  (a;^  +  5)*—  43?*  =  160,  to  find  the  values  of  x. 

Ans.  a;  =  3,  or  V  —  ^^' 


PRAXIS,  261 

13.  Given  (ar'  — 7  a;)  +  (ar^  — 7  a;  +  18)^  =  24,  to  find  the 
values  of  z.  Ans.  a;  =  9,  or  —  2. 

14.  Given  2ar^  +  3a;  — 5(2a:'  + 82;  + 9)^ +  3  =  0,  to  find 
the  values  of  a;.  Ans.  a;  =  3,  or  — 4J. 

15.  Given  a;  +  (a;  +  6)^  =  2  +  3  (a;  -f-  6)^,  to  find  the  values 
of  z.  Ans.  z  =  10,  or  —  2. 

z    I   a?^        3;  —  a; 

16.  Given       "^      =  — - — ,  to  find  the  values  of  z. 

^  —  ^^  Ans.  a;  =  4,  or  1. 

(Q\  2                                         Q 
a;  -|-  -  I  -j-  a;  =  42 ,  to  find  the  values  of  z. 

Ans.  a;  =  4,  or  2.    . 

^^*  ^^^^^  (^ITlp  +  ?:Z4 "=  25?'  *^  ^"^"^  ^^^  ^^^"^^^  °^  ^' 

Ans.  a;  =  ±  3. 
19.  Given  [(a;  — 2)''  — a;P— (a;--2)''  =  90  — a;,  to  find  the 
values  of  z.  Ans.  a;  =  6,  or  —  1 . 

^'  ^id " + ^ = f  ~  ""^  1 '» '^"'j  t'^^  ^^1"^^  °f  ^  ='"''  2'- 

From  the  first  equation  7?y^  -\-  ^zy  =  96,  completing  the  square 
and  extracting  the  root  zy  =  S,  or  — 12. 

Ans.  a;  =  4  or  6,  ?/  =  2  or  4. 

21.  Givenar^y^  — 7a;2/^  — 945  =  765^)  to  find  the  values  of 

and  zy  —  y     =  12    J  z  and  y. 

Ans.  z-==-5,  y  ==  3. 

22.  Given  ar'-[-a;-[-y=18  —  'f)  to  find  the  values   of  z 

anda;2/=6  (  and  y. 

From  the  first  equation  3?  -\'  'f  -\-  z  -\-  y  =\Q 

from  the  second  2 zy  =12 

by  addition  •      x^ -j" ^^V  +  2/'  +  2;-[-2/  =  30 

or  {x  +  yf+{z  +  y)=20 

whence  a;  =  3  or  2,  y=2  or  2. 

23.  Given  z^-\-if  —  z  —  y==78)  to  find  the  values  of  x 

and  a;y-|"Z-|-y  =  39  S  and?/. 

Ans.  a;  =  9  or  3,  y  =  3  or  9. 

24.  Given  z^  4-3 z-{-y  ==73  —  2zy)to  find  the  values  of  x 

and  y^ -^^  3 y-\-z  =  4:4:  )  and  y. 

Ans.  a;  =  4  or  16,  2/ =  5  or — 7. 


/ 

262  ELEMENTS    OF   ALGEBRA. 

25.  Given  x  —  2x^y^-{'y  =  x^—y^lto^nd  the  values  of 


^ndxi+y^  =  5  .)  ^^^^2/. 

From  the  first  equation  we  have  {x^  — y^f —  {x^  — 2/^)  =  0. 

Ans.  x  =  9,  7/  =  4. 

26.  Given  r*  +  2 z?/  +  ?/^  +  2a:  ==  120  —  2 ?/ )  to  find  the  val- 

and  a;?/  —  y^  =  S  j  ues  of  a;  and  y. 

Ans.  2/  :^  4  or  1,  a;  =  6  or  9. 

27.  Given  a:  +  42:^4-42/ =  21  4- 82/2 -i-4a:2  y2  )   to  find  a 

,      anda:i+2/i  =  6  '  )     ^^^2/- 

Ans.  a;  =  25,  y=l. 

IV. — MISCELLANEOUS    QUESTIONS. 

1.  A  farmer  has  a  stack  of  hay,  from  which  he  sells  a  quan- 
tity, which  is  to  the  quantity  remaining  in  the  proportion  of  4 
to  5.  He  then  uses  15  loads  and  finds  that  he  has  a  quantity 
left,  which  is  to  the  quantity  sold  as  1  to  2.  How  many  loads 
did  the  stack  at  first  contain  ?  Ans.  45. 

2.  A  person  engaged  to  reap  a  field  of  35  acres,  consisting 
partly  of  wheat  and  partly  of  rye.  For  every  acre  of  rye  he 
received  5  shillings ;  and  what  he  received  for  an  acre  of  wheat, 
augmented  by  one  shilling,  is  to  what  he  received  for  an  acre  of 
rye  as  7  to  3.  For  his  whole  labor  he  received  £  13.  Required 
the  number  of  acres  of  each  sort. 

Ans.  15  acres  of  wheat  and  20  of  rye. 

3.  A  person  put  out  a  certain  sum  at  interest  for  6J  years  at 
5  per  cent,  simple  interest,  and  found  that  if  he  had  put  out  the 
same  sum  for  12  years  and  9  months  at  4  per  cent,  he  would 
have  received  $  185  more.     What  was  the  sum  put  out  ? 

Ans.  $1000. 

4.  Two  persons,  A  and  B,  were  partners.  A's  money  re- 
mained in  the  firm  6  years,  and  his  gain  was  one-fourth  of  his 
principal,  and  B's  money,  which  was  £50  less  than  A's,  had 
been  in  the  firm  9  years,  when  they  dissolved  partnership,  and  it 
appeared  that  if  B  had  gained  £  6.  5s.  less,  his  gain  and  princi- 


PRAXIS.  263 

pal  would  have  been  to  A's  gain  and  principal  as  4  to  5.     What 
was  the  principal  of  each  ?  Ans.  £200  and  £  150. 

5.  The  crew  of  a  ship  consisted  of  her  complement  of  sailors 
and  a  number  of  soldiers.  Now  there  were  22  seamen  to  e very- 
three  guns  and  ten  over.  Also  the  whole  number  of  hands  was 
5  times  the  number  of  soldiers  and  guns  together.  But  after  an 
engagement,  in  which  the  slain  were  one-fourth  of  the  survivors, 
there  wanted  5  to  be  13  men  to  every  2  guns.  Required  the 
number  of  guns,  soldiers,  and  sailors. 

Ans.  90  guns,  55  soldiers,  and  670  sailors. 

6.  A  shepherd  in  time  of  war  was  plundered  by  a  party  of 
soldiers,  who  took  J  of  his  flock  and  J  of  a  she6p ;  another  party 
took  from  him  \  of  what  he  had  left  and  J  of  a  sheep ;  then  a 
third  party  took  ^  of  v/hat  now  remained  and  J  of  a  sheep. 
After  which  he  had  but  25  sheep  left.  How  many  had  he  at 
first?  Ans.  103. 

7.  A  trader  maintained  himself  for  3  years  at  the  expense  of 
$50  a  year;  and  in  each  of  those  years  augmented  that  part  of 
his  stock,  which  was  not  so  expended  by  one-third  thereof.  At 
the  end  of  the  third  year  his  original  stock  was  doubled.  What 
was  his  stock?  Ans.  $740. 

8.  When  wheat  was  5  shillings  a  bushel  and  rye  3  shillings, 
a  man  wanted  to  fill  his  sack  with  a  mixture  of  rye  and  wheat 
for  the  money  he  had  in  his  purse.  If  he  bought  7  bushels  of 
rye,  and  laid  out  the  rest  of  his  money  in  wheat,  he  would  want 
two  bushels  to  fill  his  sack ;  but  if  he  bought  6  bushels  of  wheat, 
and  filled  his  sack  with  rye,  he  would  have  6  shillings  left. 
How  must  he  lay  out  his  money  and  fill  his  sack  ? 

^s.  He  must  buy  9  bushels  of  wheat,  and  12  bushels  of  rye. 

9.  In  one  of  the  corners  of  a  garden  there  is  a  rectangular 
fish-pond,  whose  area  is  one-ninth  part  of  the  area  of  the  garden ; 
the  garden  is  rectangular  and  its  periphery  exceeds  that  of  the 
fish-pond  by  200  yards.  Also  if  the  greater  side  be  increased 
by  3  yards  and  the  other  by  5  yards,  the  garden  will  be  enlarged 


264  ELEMENTS    OF   ALGEBRA. 

by  645  square  yards.     Required  the  periphery  of  the  garden,  and 
the  length  of  each  side. 

Ans.    The  periphery  is  300  yards,  and  the 
sides  are  90  and  60  yards  respectively. 

10.  A  sets  out  express  from  C  towards  D,  and  three  hours  after- 
wards B  sets  out  from  D  towards  C,  travelling  2  miles  an  hour 
more  than  A.  When  they  meet  it  appears  that  the  distances  they 
have  travelled  are  in  the  proportion  of  13  to  15 ;  but  had  A  trav- 
elled 5  hours  less  and  B  gone  2  miles  an  hour  more,  they  would 
have  been  in  the  proportion  of  2  to  5.  How  many  miles  did  each 
go  per  hour,  and  how  many  hours  did  they  travel  before  they  met? 

Ans.  A  went  4,  and  B  6  miles  an  hour,  and 
they  travelled  10  hours  after  B  set  out. 

11.  There  is  a  number  consisting  of  two  digits,  which  being 
multiplied  by  the  digit  on  the  left  hand,  the  product  is  46 ;  but  if 
the  sum  of  the  digits  be  multiplied  by  the  same  digit,  the  product 
is  only  10.     Required  the  number.  Ans.  23. 

12.  A  detachment  of  soldiers  from  a  regiment  being  ordered  to 
march  on  a  particular  service,  each  company  furnished  four  times 
as  many  men  as  there  were  companies  in  the  regiment;  but 
these  being  found  to  be  insufficient,  dach  company  furnished  3 
more  men;  when  their  number  was  found  to  be  increased  in 
the  ratio  of  17  to  16.  How  many  companies  were  there  in  the 
regiment?  Ans.  12. 

13.  A  farmer  has-  two  cubical  stacks  of  hay.  The  side  of  one 
is  three  yards  longer  than  the  side  of  the  other ;  and  the  differ- 
ence of  their  contents  is  117  solid  yards.  Required  the  side  of 
each.  Ans.  5  and  2  yards  respectively. 

14.  A  and  B  purchased  a  farm  containing  900  acres  of  land, 
at  the  rate  of  $2  an  acre,  which  they  paid  equally  between 
them;  but  on  dividing  the  same,  A  got  that  part  of  the  farm, 
which  contained  the  best  of  the  improvements,  and  agreed  to  pay 
45  cents  an  acre  more  than  B.  How  many  acres  had  each,  and 
at  what  price  ?  Ans.  A  had  400  acres  at  $  2,25  an  acre, 

and  B  500  acres  at  $  1,80  an  acre. 


GENERAL  THEORY  OF  EQUATIONS.  S05 

SECTION  XXVI.  — General  Theory  of  Equations. 

221.  The  equations  thus  far  considered  are  of  the  first  and 
second  degrees  only.  Those  of  the  third  degree  come  next  in 
order.  We  now  proceed,  however,  to  develop  the  general  the- 
ory of  equations. 

No  general  fonnulas  can  be  given  for  the  solution  of  equa- 
tions of  a  degree  higher  than  the  fourth.  And  the  attention  of 
mathematicians  has  been  directed  chiefly  to  the  solution  of 
numerical  equations,  that  is,  to  those  which  arise  from  the  alge- 
braic translation  of  a  problem  in  which  the  given  things  are 
particular  numbers.  We  shall  give  an  elementary  view  of  the 
principles  by  means  of  which  this  object  has  been  successfully 
accomplished. 

222.  In  the  numerical  operations  required  in  the  solution  of 
equations  of  this  kind,  particularly  that  of  division,  certain  sim- 
plifications are  of  great  utility.     We  will  first  explain  them. 

DETACHED   COEFFICIENTS. 

1.  To  multiply  x^  _  Sa;^  +  3^;  —  1  by  x^  —  2a;  +  1. 
The   operation  may  be  abridged  by  first   performing   the 
multiplication  upon  the  coefficients  detached  from  the  letters, 
and  afterwards  annexing  the  letters  raised  to  the  proper  powers. 
Commencing  with  the  coefficients  the  work  will  stand  thus : 
1—3+    3—    1 
1  —  2+    1 
1  —  3+    3—    1 
_2+   6—   6  +  2 

1_   3  +  3—1 


1  —  5+10—10  +  5-1. 

The  product  of  a:^  by  a;^  is  a:* ;  the  highest  power  of  x  in  the 
product  will  be,  therefore,  ar* ;  and  since  from  the  arrangement, 
the  powers  of  this  letter  go  on  decreasing  by  unity,  we  shall 
have,  it  is  evident,  for  the  powers, 

23 


266  ELEMENTS    OF   ALGEBRA. 

Annexing  these  to  the  coefficients,  the  required  product  will 

be 

x^  —  bx^-^lOa^—lOx'^  +  bx  —  l. 

2.  If  any  powers  are  wanting  in  either  of  the  factors,  they 
must  be  supplied  by  writing  them  with  0  as  a  coefficient.  Thus, 
let  it  be  required  to  multiply 

S3^—l:i^y  +  8xy-6y^  hj  2x^ —  3xy  +  f 
Here  a  term  xi^  in  the  multiplicand  is  wanting,  which  must 
be  supplied,  thus,  Oxy^.     The  operations  upon  the  coefficients 
will  then  be  as  follows  : 

3-    7+    8+    0-    5 

2-    3+    1 

6—14  +  16+   0  —  10 
—   9  +  21  —  24—   0  +  15 

3—   7+    8+    0  —  5 
6  —  23+40  —  31—   2  +  15  —  5. 

The  powers  of  x  go  on  decreasing  by  unity,  and  those  of  y  in- 
creasing by  unity.    Supplying  these,  the  product  required  will  be 
6x^  —  23x'y-\-A0xy  —  3lxy  —  2xY  +  15xy^—5f 

3.  Multiply  6x^  —  Sax^  +  5a^x  —  a^  hy  a" -{-Sax -{- 6xK 
In  this  case  we  reverse  the  order  of  the  terms  in  the  multiplier, 
so  that  the  arrangement  may  be  the  same  as  in  the  multiplicand. 
The  operation  performed  upon  the  coefficients,  as  above,  will  give 

25  —  0  +  21  +  7+2  —  1. 
And  the  required  product  will  be 

25  a;«  +  2 1  2; V  +  7  a: V  +  2  a:  a*  -  a', 

4.  Multiply2a^  — 3fli2^53'^by2a2__5^2. 

Ans.  4:a' -^160^5'+ 10 a'b^+  15ab'-2bb'. 

5.  Multiply  x"^  —  aa^  -{-  a^x^  —  a^x  -\-  a"^  hy  x  -}-  a. 

Ans.  x^  +  a^. 

The  process  above  is  called  Multiplication  by  Detached  Co- 
efficients.  The  examples,  art.  24,  will  serve  as  an  additional 
exercise. 

223.    The   process  of   division  may,   in  like   manner,  be 


GENERAL  THEORY  OF  EQUATIONS.  267 

abridged  by  first  performing  the  operation  upon  the  coefficients 
detached  from  the  letters,  and  then  supplying  the  letters. 

1.  Let  it  be  required,  for  example,  to  divide  c^  —  ha^x-\- 
]OaV—  lOaV  +  5aa;*  —  a;^  by  a2  _  2  aa;  +  x^. 
The  operation  upon  the  coefficients  will  be  as  follows  : 

1-2+1 
1_34.3-1 


1 

1 

-5  + 
-2  + 

10- 
1 

10  +  5- 

1 

-3  + 

-3  + 

.9- 
6- 

10 
3 

3- 

3- 

7  +  5 
6  +  3 

— 

1  +  2- 
1  +  2- 

-1 
-1. 

The  coefficient  of  the  quotient  will  be  1  —  3-}-3  —  1.  And, 
m  order  to  supply  the  letters,  we  take  the  quotient  of  the  letters 
in  the  first  term  of  the  dividend  by  those  of  the  divisor ;  thus, 
a*  divided  by  c^  gives  c^.  The  letters  in  the  first  term  of  the 
quotient  will  then  be  a^,  and  in  the  succeeding  terms  they  will 
follow,  it  is  evident,  the  law  of  the  dividend.  The  quotient 
required  will  then  be  c^—  3  a^a;  +  3  a  a;^  —  a:^. 

2.  Divide  6  a%^  +  3  a%^  —  4  a^i*  +  ^«  by  3  a^Zi  -  2  a  3^  +  h\ 
The  operations  upon  the  coefficients  will  stand  thus : 


6+3-4+0+1 
6+0-4+2 

3+0-2+1 
3  +  0-2  +  1. 


3-1-0-2+1 
2+1 


Supplying  the  letters  we  shall  have  for  the  quotient,  2ah  -^-W. 

Before  commencing  the  operations,  the  dividend  and  divisor 
should,  it  is  evident,  be  arranged  both  in  reference  to  the  same 
letter.     The  process  is  called  Division  by  Detached  Coefficients. 

SYNTHETIC   DIVISION. 

224.  The  operation  for  finding  the  coefficients  of  the  quotient 
may  be  still  further  abridged. 


268  ELEMENTS   OF   ALGEBRA. 

In  the  ordinary  process  of  division,  we  multiply  the  divisor 
by  each  term  of  the  quotient  as  it  is  found,  and  subtract  suc- 
cessively the  partial  products  from  the  dividend.  The  effect, 
it  is  evident,  will  be  the  same,  if  we  change  the  signs  of  the 
divisor  and  add  the  j)artial  products  to  the  dividend.  Thus,  in 
the  first  example  above,  if  we  change  the  signs  of  the  divisor, 
and  then  find  the  terms  of  the  quotient  by  the  first  term  of  the 
divisor  with  its  sign  unchanged,  the  partial  products  may  be 
added,  and  the  work  will  stand  as  follows : 

-1+2-1 
1_-3_|-3__1 


1-5  + 

-1  +  2- 

10- 

1 

10  +  5-1 

-3  + 
3- 

9- 
6  + 

10 
3 

3- 

-3  + 

7  +  5 
6-3 



1  +  2—1 
1  —  2  +  1. 

In  this  operation  it  is  easy  to  see  that  the  terms  +  9  —  10 
in  the  second  partial  dividend,  —  7  +  5  in  the  third,  and  +  2 
—  1  in  the  fourth,  may  be  omitted ;  and  the  first  term  in  each 
partial  dividend  found  by  adding  all  the  terms  in  each  column 
as  the  Work  proceeds.  With  this  modification  the  work  will 
stand  thus : 

1-5  +  10-10  +  5-11-1  +  2-1 


-1+2-    1  I      1-3+3-1 


-3 

+  3  + 

6  + 
3 

3 

- 

-3  + 

6- 

-3 

— 

1 

+ 

1- 

-2+1 

0      0      0. 

In  this  process  there  is  liability  to  error  in  the  signs  of  the 
quotient,  in  consequence  of  the  necessity  of  finding  each  term 


GENERAL  THEORY  OF  EQUATIONS.  269 

of  the  quotient  by  means  of  the  first  term  of  the  divisor  with  its 
sign  unchanged.  To  avoid  this  liability,  recollecting  that  the 
first  term  in  each  successive  dividend  is  always  cancelled  by 
the  product  of  the  first  term  of  the  divisor  by  the  corresponding 
term  of  the  quotient,  we  retain  the  first  term  of  the  divisor  with 
its  sign  unchanged,  and  change  all  the  rest.  The  operation 
will  then  stand  thus  : 

1  —  5+10-10  +  5-1  11+2-1 


2-    1 

l__3_i_3_l 

-3 

-6+3 

-    3 

+    6-3 

-    1 

-2+1 

0      0 

The  work  may  be  written  more  concisely  thus  : 

1 

1_5_|_  10 -10  +  5-1 

2 

2_   6+   6-2 

-1 

_    1+   3-3  +  1 

First  term  of  Dividends, 

_3+   3-100 

Quotient,  1  —  3+3—1 

The  divisor  is  placed  at  the  left  of  the  dividend  in  a  vertical 

column.     Beneath,  in  a  horizontal   line,  are  placed   the   fitst 

terms  of  the  successive  partial  dividends  ;  and  under  the  whole 

is  written  the  quotient  also  in  a  horizontal  line.     The  partial 

products  are  written  under  the  tenns  of  the  dividend  to  which 

they  belong,  in  a  diagonal  line  from  the  left  downwards  toward 

the  right. 

23=* 


270  ELEMENTS    OF   ALGEBRA. 

2.  Divide  2d}  -  6^*  +  4^^  —  7^^  +  9 
by2a3  +  6a2__io. 


10 


2      0        0-6+4-7  0+9 

--6 +  18 -54+ 150 -372 


0  0        0  0  0  0 


10-   30+   90-250+620 


_  6  +  18  —  50  +  124  -  289  -  250  +  629. 


1__3+   9-25+   62. 

The  operation,  it  is  evident,  terminates  when  the  partial 
products  have  reached  the  right  hand  column.  This  is  the 
case,  in  the  present  example,  when  the  term  62  of  the  quotient 
is  obtained.  And  since  the  columns  to  the  right  of  this  do  not, 
when  added,  severally  reduce  to  0,  there  will  be  a  remainder, 
of  which  the  sums  of  these  columns  respectively  will  be  the 
coefficients. 

Supplying  the  letters,  we  shall  have,  therefore,  a^  —  3  a^  +  9 
a^  —  25  a  +  62  for  the  quotient,  with  a  remainder  —  289  c?  — 
250  a  +  629. 

3.  Divide  x^ —  hs? -\-\hx'' -2^7? -\-'n  x^ -\^x-\-h  by 
a:*  _ 2 a:^  +  4 a;2  —  2 a;  +  1 .  kx\&,:^ —  ^x-^-h. 

4.  Divide  a:*  +  2  a;*?/ +  3  a:^?/^  —  a:y  —  2  a:  ^  —  3  ?/«  by  x^ 
+  2  a:  y  +  3  2/^.  Ans.  o?  —  if. 

The  process  with  the  modification  above  is  called  Synthetic 
Division.  The  examples,  art  39,  will  furnish  an  additional 
exercise  foiH;he  learner. 

General  Properties  of  Equations. 

225.  Any  expression  which  involves  a  quantity  is  called  a 
fimction  of  that  quantity. 

Thus,  x^  -\-pXja3^-{'by  {a-\-xf  are  all  functions  of  x,. 

In  like  manner,  ax^  —  by^,  x^y  +  j/^ar,  are  functions  of  z 
and  y. 

2.  A  function  is  usually  indicated  by  some  one  of  the  letters, 
/,  F,  &c.,  the  quantity  or  quantities  of  which  the  expression  is 


GENERAL  THEORY  OF  EQUATIONS.  271 

a  function  being  inclosed  in  a  parenthesis.  Thus,/  (2:)  indicates 
a  function  of  a;,  /  {x,  y)  a  function  of  x  and  y. 

3.  If  hyf{x)  we  denote  a  particular  function  of  x,  then  f  {a) 
will  denote  the  same  function  of  a.  Thus,  if  the  first  function 
is  x^  -{-  6  X  -{-  6,  the  second  will  he  a^--\-ba-\-6. 

4.  It  will  be  recollected  that  by  the  root  of  an  equation  we 
understand  any  quantity  which,  being  substituted  in  the  equa- 
tion, will  satisfy  its  conditions.     . 

5.  An  equation  of  the  second  degree  is  sometimes  called  a 
quadratic  equation,  one  of  the  third  degree  a  cubic^  and  one  of 
the  fourth  a  bi-quadratic  equation. 

6.  A  complete  equation  of  the  nth.  degree  with  one  unknown 
quantity,  n  being  an  entire  and  positive  number,  may  be  re- 
duced to  the  form, 

a:"+Aa:"-i4-Ba:'-2  +  Ca;"-3-|-     .     .     .     Ta:  +  U  =  0,in 

which  the  coefficients  A,  B,  C  .  .  .  T,  U,  are  any  num- 
bers whatever,  positive  or  negative,  entire  or  fractional. 

Every  equation  of  this  description,  since  it  is  supposed  to  be 
derived  from  a  problem  with  sufficient  and  properly  limited 
conditions,  may  be  assumed  to  have  at  least  one  root. 

We  now  proceed  to  investigate  the  general  principles  neces- 
sary to  the  solution  of  numerical  equations  of  any  degree. 

Divisibility  of  Equations. 

226.  1.  Resuming  the  general  equation, 
a:«  +  Aa;"-i  +  Ba;'-2+     .     .     .     Ta:  +  U=0,  (1) 
if  a  is  a  root  of  the  equation,  then  the  jirst  member  is  divisible 
by  X  —  a. 

For  if  the  division  is  not  exact,  let  Q  be  the  quotient,  and  R 
the  remainder  arising  from  the  division  by  x  —  a ;  then  we 
have 
a:''  +  Aa;"-i+    ....    T  a:  +  U  =  Q  (2:  -  c) +  R.     (2) 

But  the  left  hand  member  of  this  equation  is  equal  to  0  ;  and 


272  ELEMENTS    OF    ALGEBRA. 

since  a  is  by  hypothesis  a  root  of  the  equation,  we  have  a;  =  a, 
or  »  —  a  =  0,  and  the  equation  (2)  reduces  to 

0=:0  +  R,  orR  =  0, 
that  is,  there  is  no  remainder,  and  the  division  is  exact. 

2.  Conversely,  if  the  first  member  of  the  equation  (1)  is  divisi- 
ble by  a;  —  «,  then  a  is  a  root  of  the  equation.  For  Q  being 
the  quotient  arising  from  the  division  by  a;  —  «,  the  equation 
returns  to  Q  (a:  —  a)  =  0, 

which  is  satisfied  by  the  value  x  =  a\  hence  a  is  a  root  of  the 
equation. 

In  the  solution  of  equations  we  have  frequent  occasion  to 
ascertain,  by  trial,  whether  a  particular  number  is  a  root  of  the 
equation.  From  the  preceding  principle  it  is  obvious  that  this 
may  easily  be  done  by  division. 

Ex.  1.  To  determine  whether  4  is  a  root  of  the  equation, 
a:3_9^2_^26a:-24  =  0. 

Dividing  by  z  —  4,  and  performing  the  operation  by  synthetic 
division  we  have 


1-9  +  26  —  24 

4_20  +  24 


1-5+   6 
Ans.  4  is  a  root,  and  if  the  proposed  be  divided  by  a;  ~  4 
the  equation  which  results  will  be 

Ex.  2.  To  determine  whether  5  is  a  root  of  the  same  equa- 
tion. 

Ans.  5  is  not  a  root,  since  the  division  by  a;  —  5  leaves  a 
remainder  of  6. 

Ex.  3.  Is  2  a  root  of  the  equation  a;^  —  7a;  +  6  =  9? 

Ex.  4.  Is  3  a  root  of  the  equation  s^-^Qx^  +  Sx'-l^ 
=  0? 


GENERAL  THEOEY  OF  EQUATIONS.  273 


Number  of  the  Roots. 

•  227.  In  order  to  the  solution  of  an  equation,  we  must  first 
determine  the  number  of  its  roots.  An  equation  of  the  second 
degree  with  one  unknown  quantity  has,  we  have  seen,  two 
roots.  We  shall  now  show  that  every  equation  with  one 
unknown  quantity  has  as  many  roots  as  there  are  units  in  the 
highest  power  of  the  unknown  quantity,  and  no  more. 
Let  a  be  a  root  of  the  equation 

x-J^Ax^-^  +  Bx^-^^    .     .     .     Tx-\-V  =  0; 
since  by  the  last  article  this  equation  is  divisible  by  a;  —  a,  it 
returns  to 

{x-a)  (a;'-i  +  AV-'^+     .     .     .     Tx-\-V\)=:0, 
A',  &c.,  being  the  new  coefficients  which  arise  from  the  division. 
But  this  equation  is  satisfied  by  a:  —  a  =  0,  or  by 

a;"-i  +  AV-2+     ....     Tx  +  V'  =  0. 
Let  3  be  a  root  of  this  last  equation,  then  we  have 
(a:-i)(a;"-2  +  A'V-3+     .     .     .     T"a:  +  U'0  =  O, 
which  is  satisfied  by  a:  —  3  =  0,  or  by 

x^^  +  A"x^-i-  ....  T''a:+U'  =  0. 
Continuing  the  operation,  it  will  be  seen  that  for  every  new 
factor  obtained,  the  exponent  of  x  is  made  one  less,  and  that  we 
shall  have  finally  a:"  +  A  a;"-!  4- Ba;''-2+  ,  .  .  Tx  +  U 
e={x  —  a){x  —  b)  {x-^  c)  .  .  .  .  {x  — p)j  in  which  the 
number  of  binomial  factors,  x  —  a,  x  —  i,  &c.,  is  equal  to  n  or 
to  the  number  of  units  in  the  index  of  the  highest  power  of  the 
unknown  quantity.  And  since  there  are  as  many  roots  as 
factors,  there  will  be  as  many  roots  as  units  in  the  highest 
power  of  Xf  the  unknown  quantity. 

An  equation,  moreover,  cannot  have  a  greater  number  of 
roots  than  there  are  units  in  the  highest  power  of  x. 

LetV=a;''  +  Aa;'-i  +  Ba;'-2+   ....   Ta:  +  U  =  0, 
the  roots  of  which  are  a,  i,  c    .     .     .     .    ^,  respectively ;  then 

Y  =  (x-'a){x—b){z-.c) {x-ph 

18 


274  ELEMENTS   OF   ALGEBRA. 

If  it  be  possible,  let  a!  be  another  root  differing  frpm  a,  3,  c, 
.     .•    .    p\  then  we  shall  have 

\^{a' -a){a! -h){d  —  c)  .    ,    .    .    («'_j9)=0; 
but  this  equation  is  impossible,  since  a'  being  different  from  a, 
3,  c,    .     .    .    .    ^,  no  one  of  the  factors  of  V  can  be  equal  to  0. 

'Every  equation^  therefore^  will  have  as  manij  roots  as  there  are 
units  in  the  highest  power  of  the  unknxmn  quantity,  and  nx)  more. 

These  roots  may  not,  however,  be  all  different.  In  fact,  any 
number  of  them  may  be  equal,  as  a  and  3,  or  a,  h,  and  c,  &c. 

If  the  equation  has  two  roots,  each  equal  to  a,  for  example,  it 
will  be  divisible  by  {x  —  aY;  if  it  has  three  roots,  each  equal  to 
fl,  it  will  then  be  divisible  by  {x  —  of,  and  so  on. 

A  part  of  the  roots,  moreover,  may  be  imaginary.  But,  from 
what  has  been  said,  every  equation  will  have  at  least  one  real 
root. 

228.  From  what  has  been  done,  it' will  be  seen  that  if  one  or 
more  roots  of  an  equation  are  known,  the  reduced  equation  con- 
taining the  other  roots  may  easily  be  found  by  division. 

Ex.  1.  One  root  of  the  equation  x^ — \bx'^-\-1bx  — 125 
=  0,  is  5.  What  is  the  equation  which  contains  the  other 
roots? 


By  Synthetic  Division,  1 
5 


1^15  +  75_125 
5_50-Ll25 


1  _  10  +  25. 

Ex.  2.  Two  roots  of  the  equation  a:'*  —  ba^—\2x'^-\-lQx 
—  80  =  0,  are  2  and  5.  What  is  the  reduced  equation  which 
contains  the  other  roots  1 

Operation. 


1st  Division,  1 
2 

2d  Division,  1 
5 


1__5__  12  +  76  — 80 
2—   6  —  364-80 


1  —  3  _  18  +  40 
5+10  —  40 


1+2—   8 

Ans.  a;2  +  2a:  — 8  =  0. 


GENERAL  THEORY  OF  EQUATIONS. 


275 


If,  as  in  the  preceding  example,  the  reduced  equation  is  a 
quadratic,  the  remaining  roots  may  be  found  by  the  methods 
already  explained. 

Ex.  3.  One  root  of  the  equation  a^  +  3x^— I6x+I2=z0 
IS  1 ;  what  are  the  remaining  roots  ?  Ans.  2,  and  —  6. 

Ex.  4.  Two  roots  of  the  equation  x^  —  12a^ -\-48x^ —  G8z 
-|-  15  =  0,  are  3  and  5.    What  are  the  remaining  roots  ? 

Ans.  2  +  V  3,  and  2  —  V  3. 

Ex.  5.  One  root  of  the  equation  a^  —  x^  —  7  a;  -f-  15  =  0,  is 
—  3,     What  are  the  other  two  roots  ? 

Ans.  2  +  a/  —  1,  and  2  —  yy/  —  1' 


Coefficients. 

229.  The  roots  of  an  equation  are  obviously  involved  in  the 
coefficients.  We  proceed  next  to  determine  the  law  of  the 
coefficients,  or  the  manner  in  which  they  are  connected  with 
the  roots. 

Let  it  be  proposed,  then,  to  form  the  equation  whose  roots 
shall  he  a,  b,  c    .    .    .    .    respectively. 

The  left  hand  member  will  be  equal,  it  is  evident,  to  the  con- 


tinued product  of  X  —  a,  X  —  bj  x 
the  multiplication  we  have 

{x  —  a)  {x  —  b)  -zzzT?  — 

{x  —  a)  (a:  —  b)  {x 


x-\'ah 


Performing 


abc 


c)=^a^  —  a    a^-^ab 
—  b  ac 

•— c  be 

and  so  on,  as  in  art.  128. 

From  what  has  been  done  we  have  the  following  properties, 
yviz. : 

1^.  The  coefficient  of  the  second  term  in  the  required  equa- 
tion will  be  the  sum  of  all  the  roots  with  their  signs  changed. 

2®.  The  coefficient  of  the  third  term  will  be  the  sum  of  the 
products  of  every  two  roots  with  their  signs  changed. 


270  ELEMENTS   OF   ALGEBRA. 

3*'.  The  coefficient  of  the  fourth  term  will  be  the  sum  of  the 
products  of  every  three  roots  with  their  signs  changed,  and  so  on, 

4°.  The  last,  or  absolute  terrri  will  be  the  jn'odvx^t  of  all  the 
roots  with  their  signs  changed. 

That  this  law  is  general  may  be  shown,  as  in  art.  129. 

From  these  principles,  it  follows, 

1°.  If  the  coefficient  of  the  second  term  in  any  equation  is  0> 
that  is,  if  the  second  term  is  wanting,  the  sum  of  the  positive 
roots  is  equal  to  the  sum  of  the  negative  roots. 

2°.  If  the  signs  of  the  terms  of  the  equation  are  all  positive, 
the  roots  are  all  negative ;  and  if  the  signs  are  alternately  posi- 
tive and  negative,  the  roots  are  all  positive. 

,3°.  Every  root  of  an  equation  is  a  divisor  of  the  last  or 
absolute  term. 

1.  The  following  examples  exhibit  the  manner  in  which  the 
coefficients  are  derived  from  the  roots. 

Ex.  1.  Find  the  equation  whose  roots  are  2,  3,  4  and  — 5- 
Indicating  the  equation  it  will  be 

{X  —  2)  (a;  —  3)  (a;—  4)  (a;  +  5)  =  0. 

The  coefficients  may  be  found  by  the  principles  just  demon- 
strated, or  by  actual  multiplication  as  follows, 

1  —  2 

—  3  +  6 


—  3 

—  4 


5+  a 

4-f  20—   24 


1_9_|_26—   24 

5  —  45+130  —  120. 


1_4_19_|.106_120. 
Ans.  a:*  — 4 a:3_  19^2^  106  a;— 120  =  0. 
Ex.  2.  What  is  the  equation  whose  roots  are  1,  3,  and  —  4? 

Ans.  a:»  — 13a;+12  =  0. 
Ex.  3.  What  is  the  equation  whose  roots  are  —  1,  —  2,  —  3> 
and— 5? 

Ans.  a:*+ll«»  +  41x2  +  61a;+30  =  a 


GENERAL  THEORY  OF  EQUATIONS.  277 

Ex.  4.  Find  the  equation  whose  roots  are  2,  3,  5,  and  6. 
Ans.  ai'—16u^-\-91x'  —  216x-\-l80  =  0, 

2.  By  means  of  the  first  of  the  preceding  principles  one  of 
the  roots  of  an  equation  may  be  found,  when  all  the  rest  are 
determined. 

By  means  of  the  fourth,  the  integral  roots  may  all  b^  found. 
In  order  to  this,  we  seek  among  the  divisors  of  the  last  term 
those  that  will  satisfy  the  equation. 

Ex.  1 .  Find  the  integral  roots  of  the  equation  x^ —  8  a:^  -j- 19  a: 
—  12  =  0.  The  divisors  of  the  last  term  are  1,  2,  3,  4,  and  6 ; 
of  these  1,  3,  and  4,  substituted  respectively  for  x,  satisfy  the 
equation,  and  are,  therefore,  roots.  The  equation  being  of  the 
third  degree  only,  they  are  all  the  roots. 

Ex.  2.  Find  the  roots  of  the  equation  a^  —  2z^ — bx-\-6 
=  0.  Ans.  1,  3,  and  — 2. 

Ex.  3.  Find  the  roots  of  the  equation  a^  —  x  —  6  =  0. 

Ans.  2  is  the  only  integral  root.  Depressing  the  equation  by 
this  root,  the  remaining  roots  found  from  the  resulting  equation 
are  —  1±  V  —  2. 

FORM   OF   THE   ROOTS, 

230.  The  roots  of  an  equation  may  be  entire,  fractional,  surd 
or  imaginary. 

Let  there  be  the  equation 

a;--f  Az^-i-f  Ba:'-2+ Ta:  +  U  =  0, 

in  which  the  coefficient  of  the  first  term  is  unity,  and  A,  B,  &;c., 
entire  numbers.     To  deteraiine  whether  this  equation  can  have 

a  fractional  root : 

a      ' 
If  it  be  possible,  let  the  fraction-,  the  terms  of  which  are 

prime  to  each  other,  be  a  root  of  this  equation. 

Substituting  -  for  a:,  multiplying  both  members  by  3"~S  and 

transposing  we  obtain 

f=-Aa'-i-Ba'-2*-  ....   Tab'^'  —  Vb'^K 
^  24       / 


278  ELEMENTS   OF  AtGfiBRA. 

The  right  hand  member  of  this  equation  is  an  entire  quantity, 
since  it  is  composed  of  terms  each  of  which  is  integral.  The 
left  hand  member  must,  therefore,  be  entire,  or  we  shall  have  a 
whole  number  equal  to  a  fraction,  which  is  absurd.  Hence  an 
equation,  whose  coefficients  are  all  integers  and  that  of  the  highest 
power  of  the  unknown  quantity  equal  to  uniti/,  cannot  have  a 
fractional  root. 

It  does  not  follow  from  this,  however,  that  all  the  roots  are 
whole  numbers.  The  equation  may  have  other  roots,  which  can- 
not be  expressed  in  whole  numbers  or  definite  fractions,  such  as 
surds  or  imaginary  quantities. 

231.  But  surds  and  impossible  roots  enter  equatimis  by  pairs ^ 
so  that  if  there  be  one,  there  will  necessarily  be  two ;  and  if 
three,  there  will  necessarily  be  four,  and  so  on. 

Let  a  +  V  —-  5,  for  example,  be  one  of  the  roots  of  an  equa- 
tion, the  coefficients  of  which  are  real. 

Suppose  the  equation  reduced  by  division  until  two  only  of 
its  roots  remain.  It  will  then  be  a  quadratic.  And  if  one  of 
its  roots  \s  a  -{■  /s/  b,  the  other  will  necessarily  he  a  —  a/  —  b. 
In  the  same  way  it  may  be  shown  that  if  there  are  three  surd 
or  imaginary  roots,  there  will  necessarily  be  four,  and  so  on. 

From  this  it  follows, 

1°.  An  equation  of  an  even  degree  may  have  all  its  roots 
imaginary;  but  if  they  are  not  all  imaginary,  two  of  them,  at 
least,  will  be  real. 

2^.  The  product  of  every  pair  of  imaginary  roots  being  of  the 
form,  a^  -j-  b,  is  positive ;  hence  the  absolute  term  of  an  equation 
whose  roots  are  all  imaginary  must  be  positive. 

3°.  Every  equation  of  an  odd  degree  has  at  least  one  real 
root ;  and  if  there  be  but  one,  that  root  must  necessarily  have  a 
contrary  sign  to  that  of  the  last  term. 

4°.  Every  equation  of  an  even  degree  whose  last  term  is 
negative  has,  at  least,  two  real  roots ;  and  if  there  be  hoA  two, 
one  of  these  will  be  positive  and  the  other  negative. 

These  principles  are  illustrated  in  the  following  examples. 


GENERAL  THEORY  OF  EQUATIONS.  279 

In  forming  the  equations,  the  most  convenient  process  will  be  to 
multiply  together,  in  pairs  the  factors  containing  the  imaginary 
roots,  and  then  combine  the  factors  thus  obtained. 

Ex.  1.  Form  the  equation  whose  roots  are  2  -}-  V  —  3,  2 

—  V  —3,  3  -f-  V  —  1.  and  3  —  V  —  1- 

Ans.  a;*— 10a;3  +  41a;2  — 82a:  +  70  =  0. 

Ex.  2.  Form  the  equation  whose  roots  are  3  -f-  v^  —  5, 3  — 
a/— 5,and5.  Ans.  3^ -^  12 x^ -\- bO x  —  84.  =  0. 

Ex.  3.  Form  the  equation  whose  roots  are  5  -}-  V  —  1j  and 
5  ~  V  -  1-  Ans.  x^ —  10  X +  26  =  0. 

Ex.  4.  Form  the  equation  whose  roots  are  2,  3  -|-  V  —  4, 
3  _  V  -  4,  and  -  5. 

Ans.  a:*-32:3_  15^2^99  2.  _  130^0. 

2.  An  equation  which  has  imaginary  roots  is  divisible  by 
(x  —  a  +  b/sZ-^l)  {x  —  a  —  bA/  —  1),  or,  {x  —  af  +  b^;  a 
-\-b  /s/  —l^a  —  b/s/  —  1,  representing  any  pair  of  the  roots ; 
hence 

1°.  Every  equation  may  be  resolved  into  rational  factors, 
simple  or  quadratic. 

From  what  has  been  done,  it  is  also  evident  that, 

2°.  An  algebraic  equation  which  has  real  coefficients  is 
alw^ays  composed  of  as  many  real  factors  of  the  first  degree  as 
it  has  real  roots,  and  of  as  many  real  factors  of  the  second 
degree  as  it  has  pairs  of  imaginary  roots. 

Ex,  Form  the  equation  whose  roots  are  3,  —  5, 1  -}-  V  —  3, 
1  -  V  —3.  Ans.  x"^  -  15a;2  +  38a;  -  60  =  0. 

1.  What  are  the  factors  corresponding  to  the  real  roots  of 
this  equation  ?  Ans.  a:  —  3,  and  a:  -J-  5. 

2.  What  is  the  factor  corresponding  to  the  pair  of  imaginary 
roots  ?  Ans.  a:^  —  2  a;  -f-  4. 

Relation  of  the  Signs  to  the  Roots. 

232.  In  the  preceding  article  we  have  seen  the  important 
relation  between  the  sign  of  the  absolute  term  of  an  equation 
and  the  form  and  number  of  the  roots.     Let  us  now  examine 


280  ELEMENTS    OF   ALGEBRA. 

the  relation  of  the  signs  of  the  terms  generally  to  the  roots. 
Resuming  the  general  equation, 

a;''  +  Aa;"-i  +  Ba;''-2+  ....  Ta;  +  U  =  a,  (1) 
and  changing  the  signs  of  tke  alternate  terms  it  becomes 

a:"  — Aa:"-i  +  Ba:"-2—  ....  dbTa;:q=U  =  0;  (2)  or 
changing  all  the  signs  in  this  last,  which  will  leave  the  equation 
identically  the  same, 

-a;"  +  Aa:"-i  — Ba:"-2_|_    ....  zp  T  a:  ±  U  =  0,  (3). 

Now  if  a  be  substituted  for  x  in  equation  (1),  and  —  «  be  sub- 
stituted for  X  in  (2)  when  n  is  an  even  number,  or  in  (3)  when 
n  is  an  odd  number,  the  equations  which  result  will  be  identi- 
cally the  same.  If  then  a  is  a  root  of  equation  (1)  this  equation 
will  be  verified  by  this  substitution.  Hence  the  equation  (2), 
or  (3)  as  the  case  may  be,  will  be  verified  by  the  substitution  of 
—  a  for  a;,  and,  therefore,  —  a  is  a  root  of  the  equations  (2) 
and  (3). 

If  J  therefore^  the  signs  of  the  alternate  terms  in  an  equation 
are  changed,  the  signs  of  all  the  roots  will  be  changed, 

Ex.  1.  Form  the  equations  whose  roots  are  1,  2, 3 ;  and  —  1, 
-2,-3. 

Ans.  The  equations  are  x^  — 6a:2  -j-  11  a:  —  6  =  0,  ^^^^  ^ 
'^Qx'-^-  lla;  +  6  =  0. 

Ex.  2.  The  roots  of  the  equation  x'^ -\- 7?  —  \^ x"^  -\- l\  x -\- 
30  =  0,  are  —  1, 2, 3,  and  —  5.  What  will  be  the  roots  if  the 
signs  of  the  alternate  terms  are  changed  ? 

Since  the  negative  roots  may  be  changed  into  positive  by 
simply  changing  the  signs  of  the  alternate  terms,  the  finding  the 
real  roots  of  an  equation  is  reduced,  by  the  preceding  principle, 
to  finding  positive  roots  only. 

233.  When  in  an  equation  the  signs  continue  the  same  from 
one  term  to  the  next  following,  there  is  said  to  be  a  permanence 
of  signs  ;  and  when  the  signs  change  from  one  term  to  the  next 
following,  a  variation  of  signs.  Thus,  in  the  equation,  x'^  —  3 
r^  —  15  o:^  -f-  49  a:  —  12  =  0,  there  is  one  permanence  and 
three  variations  of  signs. 


•  GENERAL  THEORY  OF  EQUATIONS.  281 

Let  -) — I 1 1 — |-  -f-  be  the  order  of  signs  in  any 

equation,  and  let  us  introduce  into  this  equation  a  new  positive 
root  a.  In  order  to  this,  we  multiply  the  equation  by  a;  —  a. 
The  operation,  so  far  as  the  signs  are  concerned,  will  be  as 
follows : 

+  - 

— +-+ 

+±-+-+±±- 

in  which  the  ambiguous  sign  db  indicates  that  the  sign  may  be 
+  or  — ,  according  to  the  relative  magnitude  of  the  partial 
products  with  contrary  signs  united  in  the  terms  to  which  it 
corresponds. 

If  now  this  result  be  examined  with  attention,  it  will  be  seen 
that  each  permanence  is  changed  into  an  ambiguity  by  the  in- 
troduction of  the  new  positive  root  +  ^'  It  follows,  therefore, 
that  the  permanences,  take  the  ambiguous  sign  as  we  may, 
cannot  be  increased  in  the  final  product  by  the  introduction  of 
the  new  positive  root;  but,  as  the  number  of  signs  is  increased 
by  owe,  the  number  of  variations  must  be  increased  by  one. 

In  the  equation  x  —  a  =  0,  there  is  one  positive  root,  and 
one  variation.  And  since,  from  the  reasoning  above,  the  intro- 
duction of  each  new  positive  root  in  this  equation  will  produce 
at  least  one  variation,  it  follows  that  the  number  of  positive 
roots  in  any  equation  can  never  be  greater  than  the  number  of 
variations  of  sign. 

By  a  process  altogether  similar,  it  may  be  shown  that  the 
introduction  of  a  new  negative  root  produces  at  least  one  new 
permanence,  and  that  the  number  of  negative  roots  can  never 
be  greater  than  the  number  of  permanences.  Hence,  generally, 
in  a  complete  equation  of  any  degree,  the  number  of  'positive 
roots  cannjot  he  greater  than  the  number  of  variations  of  sign^ 
nor  the  number  of  negative  roots  greater  than  the  number  of 
permanences. 

24* 


282  ELEMENTS   OF   ALGEBRA. 

Ex.  1.  How  many  permanences  and  variations  of  sign  are 
there  in  the  equation  v/hose  roots  are  2,  3,  and  —  4  ? 

The  equation  x  —  2  =  0  gives  one  variation  corresponding 
to  the  positive  root  2.  If  we  now  multiply  this  by  a;  —  3,  the 
result,  x^  —  5  X  -j-  6  =  0,  gives  an  additional  variation  corre- 
sponding to  the  new  positive  root  -\-  3.  Multiplying  again  by 
2  +  4,  the  result,  a:^  —  a;^  —  14  a;  -|-  24  =  0,  gives,  as  before, 
two  variations  corresponding  to  the  positive  roots  2,  and  3,  and 
one  permanence  corresponding  to  the  negative  root  —  4. 

Ex.  2.  How  many  permanences  and  variations  in  the  equa- 
tion whose  roots  are  2,  —  3,  and  —  5  ? 

Ans.  The  equation  is  a:^  -[-  6  a;^  —  a;  —  30  =  0,  exhibiting 
one  variation  and  two  permanences. 

The  whole  number  of-  permanences  and  variations  taken 
together,  it  is  evident,  will  be  equal  to  the  degree  of  the  equa- 
tion. If  all  the  roots,  therefore,  are  real,  the  number  of  positive 
roots  will  be  equal  to  the  number  of  variations,  and  the  number 
of  negative  roots  will  be  equal  to  the  number  of  permanences. 

234.  In  what  precedes,  the  equation  is  supposed  to  be  com- 
plete. If  there  are  missing  terms  their  place  must  be  supplied 
with  the  coefficient  0.  Any  sign  may  be  given  to  this  coeffi- 
cient without  affecting  the  roots  of  the  equation. 

Let  there  be  the  equation  a;*  —  5  a:^  -|-  8  a:  —  6  c=  0.  In  its 
present  form  this  equation  exhibits  variations  only.  It  would 
appear,  therefore,  that  it  can  have  no  negative  roots.  But  if 
we  supply  the  missing  term  it  becomes 

a;4  _j_  0  a:3  _  5^_|_  3^.  _  g  ^  0. 

Now  which  ever  sign  we  take  with  the  0  coefficient,  the 
equation  will  present  one  permanence.  It  may  have,  therefore, 
one  negative  root. 

The  preceding  principle  enables  us  to  determine  the  number 
of  positive  and  negative  roots  a  proposed  equation  may  have, 
an  object  of  importance  in  the  research  for  the  roots. 

Ex.  1.  The  equation  a;*^  — 3a:4  — 5a;^-f  15a:2^4a._12 
=  0,  has  five  real  roots.     How  many  of  them  are  positive  ? 


GENERAL  THEORY  OF  EQUATIONS.  283 

Ex.  2.  The  equation,  a:^  —  3  a:'  —  15  a;^  -j-  49  a;  —  12  =  0, 
has  four  real  roots.     How  many  of  them  are  negative  ? 

Ex.  3.  The  equation  a:*  —  3a;' —  15a;2+ 99a:— 130=0, 
has  (art.  231)  two  real  roots.     What  are  their  signs  ? 

235.  By  means  of  the  above  rule  we  are  sometimes  also 
enabled,  when  there  are  missing  terms,  to  detect  the  existence 
of  imaginary  roots  in  an  equation. 

Thus,  let  there  be  the  equation  a:^  _|_  30  =  0. 

Supplying  the  missing  term,  it  becomes 
a:2±0a;  +  30  =  0. 
-  If  the  upper  sign  be  taken  with  the  coefficient  0,  there  will  be 
no  variations ;  hence  there  can  be  no  positive  roots.  And  if  the 
lower  sign  be  taken,  there  will  be  no  permanences ;  hence  there 
can  be  no  negative  roots.  The  two  roots  of  the  equation  will 
therefore  be  imaginary. 

Let  there  be  next  the  equation  a:^  -)-  10  a;  -f-  5  =  0. 

Supplying  the  missing  term,  it  becomes 

a:3_j_0a;2_f-l0a;-|:5  =  0. 
Here,  if  the  upper  sign  of  the  0  coefficient  be  taken,  the  equation 
exhibits  only  permanences ;  it  can  have,  therefore,  no  positive 
root.  If  the  lower  sign  be  taken  there  will  be  but  one  per- 
manence, and,  therefore,  there  can  be  but  one  negative  root. 
Thus  it  appears  that  the  equation  must  have  two  imaginary 
roots. 

Transformation  of  Equations. 

236.  When  the  solution  of  an  equation  is  difficult,  the  work 
may  sometimes  be  accomplished  with  more  facility  by  aid  of 
another  equation,  the  roots  of  which  shall  bear  to  those  of  the 
proposed  some  given  relation.  The  roots  of  the  latter  being 
found,  those  of  the  former  will  then  be  readily  deduced  from 
them. 

It  may,  therefore,  be  required  to  change  a  proposed  equation 
into  another,  the  roots  of  which  shall  bear  to  those  of  the  former 
a  given  relation.  One  of  the  most  simple,  as  well  as  useful,  of 
=  0? 


284  ELEMENTS   OF   ALGEBRA. 

these  transformations  is  to  change  the  equation  into  another, 
the  roots  of  which  shall  be  greater  or  less  than  those  of  the  pro- 
posed by  a  given  quantity. 

To  effect  the  transformation  required,  it  will  be  sufficient, 
it  is  evident,  to  substitute  for  x  in  the  proposed  x  diminished  or 
increased  by  the  desired  quantity.  The  resulting  equation  will 
be  of  the  same  form  with  the  proposed,  the  roots  of  which  will 
be  of  the  required  dimensions. 

Let  it  be  required,  for  example,  to  find  an  equation  whose 
roots  shall  be  less  by  2  than  those  of  the  equation 

a:2_8a;-|-7  =  0.     (1) 

Substituting  a;  -|-  2  for  x^  the  equation  becomes 

(2:  + 2)2- 8  (a; +  2) +  7  =  0; 

or  developing  and  reducing, 

a;2_4a:  — 5  =  0.  (2) 

Resolving  equation  (1)  its  roots  are  7  and  1.  Resolving 
equation  (2)  its  roots  are  5  and  —  1,  or  less  by  2  than  those  of 
the  proposed,  as  was  required. 

Ex.  2.  Let  it  be  required  to  find  next  an  equation  whose 
roots  shall  be  greater  by  2  than  those  of  the  equation 

Ans.  a:^— 12a:2  +  47a:  — 60  =  0. 
237.  The  process  above  is  laborious,  especially  in  the  higher 
equations.     Let  us  see  if  one  more  simple  can  be  found. 
Resuming  the  general  equation 

a:"  +  Aa;'^i+Ba;"-2-l-    .     .     .     .     Ta:  +  U=0, 
let  us  make  xz=.u-\-x\u  being  a  new  unknown,  and  x'  an 
indeterminate,  to  which  any  value,  positive  or  negative,  may  be 
assigned  at  pleasure. 

Substituting  u  -f  x\  or,  which  is  the  same  thing,  x!  •\'uiox  x 
in  the  proposed,  it  becomes 

(a;'  +  2^)"-f  A(a:'  +  w)'-i-|-B(a;'+t^)'-?  .  .  .  .T(a;'+M) 
+  U==0,(1) 


GENERAL  THEORY  OP  EQUATIONS. 


285 


or  developing  the  several  terms  by  the  binominal  formula,  and 
arranging  with. reference  to  the  ascending  powers  of  m,  . 


a:" 
U 


u  + 


1 


1.2 


u^  + 


....    M''  =  0.  (2) 

Observing  the  manner  in  which  the  different  coefficients  of  u 
are  formed,  the  following  remarkable  law  will  be  discovered  : 

P.  The  coefficient  of  u°  is  simply  what  the  first  equation 
becomes  when  x'  is  substituted  for  x. 

Let  us  designate  this  coefficient  by  X. 

2°.  The  coefficient  of  u  is  formed  by  means  of  the  preced- 
ing or  X,  by  multiplying  each  of  the  terms  of  X  by  the  exponent 
of  x'  in  that  term,  and  diminishing  this  exponent  by  unity. 

Denote  this  coefficient  by  Y. 

3°.  The  coefficient  of  w^  is  formed  from  Y,  by  multiplying 
each  term  of  Y  by  the  exponent  of  x'  in  that  term,  dividing  the 
product  by  2,  and  diminishing  each  exponent  by  unity. 

Z 

Let  —  be  this  coefficient.     It  is  evident  that  Z  is  formed  in 
di 

the  same  manner  from  Y  that  Y  is  formed  from  X.     In  general 

the  coefficient  of  any  power  of  u  is  formed  from  the  preceding 

coefficient  in  the  following  manner,  viz. : 

By  taking  each  term  of  the  •preceding  coefficient  in  succession, 
multiplying  it  by  the  exponent  of  x\  dividing  by  the  number 
which  maj'ks  the  place  of  the  coefficient,  and  diminishing  the  ex- 
ponent  of  x'  by  unity. 

The  preceding  development  will  be  represented  generally  by 

X  +  Yu  +  ^u'+^u'  +  ko. 

The  polynomials,  Y,  Z,  &c.,  are  called  derived  polynomials  of 
X,  since  Z  is  derived  in  the  same  manner  from  Y,  that  Y  is 


286  ELEMENTS   OF   ALGEBRA. 

from  X.     Y   is  called  the  firsts  Z  the  second,  V  the  third, 
derived  polynomial  of  X,  and  so  on. 

Ex.  Let  X  =  a;3  _  6  x^  -j-  H  a:  —  6  =  0.  To  find  the 
derived  polynomials. 

Ans.  Y  =  32;2-  12a;  +11 
Z=6a;  -12 
V  =  6. 

238.  Returning  to  our  purpose,  let  it  now  be  proposed  to  find 
an  equation,  the  roots  of  which  shall  be  greater  by  unity  than 
those  of  the  equation 

4a;3_5a;2_|-7a:  — 9  =  0. 
Let  us  put  x=:u — 1,  from  which  we   have  u=.x-\-\. 
From  the  preceding  development  the  transformed  equation  will 
be  Z  V 

To  find  the  coefficients,  we  have  therefore 
5(_l)2-l-7(— 1)  _9  =  — 25 
10  (—  1)  4-  7  =      29 

5  ==  —  17 

=        4 

The  transformed  required  will  be 

4w«  —  17  ^2  _(_  29  w  -  25  =  a 
Ex.  2.  Transform   the   equation   a:*  — 4a;^  — 8a:4-32  =  0 
into  another,  whose  roots  shall  be  2  less. 

Ans.  a:*-[-4a;^  —  24  a;  =  0. 

239.  The  process  above  being  still  laborious,  another  more 
simple  is  to  be  sought. 

In  order  to  this,  we  resume  the  general  equation,  and  also 
equation  (2)  art.  237,  denoting  the  coefficients  of  this  last  by 
AVB'   ....     TMJ',  respectively,  thus, 

af  +  Aa;'^^  +  B2:'-2+ T^_l_U^O,     (1) 

M-  +  A'M"-^  +  B'w"-2_j_  ^    ....   T'w  +  U'  =  0.  (2) 
Substituting  in  this  last  for  u  its  value  x  —  x\  we  obtain 


in 

which 

2;'  =  - 

-L    T 

X  = 

=   4(- 

-\f- 

Y  = 

:12(- 

-\f- 

Z 

2  "" 

12  (- 

-If- 

V 
2.3  = 

=  4 

GENERAL  THEORY  OF  EQUATIONS.  287 

^x  —  afy-\-A'{x-xy-^  +  B'{x-xy-^-\-  ....  T' (a; -a;') 
+  F  =  0,  (3) 

which  when  developed  must,  it  is  evident,  be  identical  with 
equation  (1) ;  since  the  second  is  formed  from  the  first  by  the 
substitution  of  u-\-x'  for  x,  and  the  third  is  formed  from  the 
second  by  substituting  x  —  x'  for  u,  by  which  we  necessarily 
return  to  (1),  the  original  equation.  We  have,  therefore, 
^x  —  x'y  +  A'ix  —  xT-^-^  .  .  .  T{x  —  x')  +  V'  =  x'' 
+  Aa;'^i+    .     .     .     Ta:  +  U.     (4) 

Now,  if  we  divide  the  first  member  of  this  equation  by  a;  —  x^, 
ever)'-  term  will  be  divisible  by  it,  except  the  last,  which  will  be 
the  remainder  after  the  division.  And  since  the  two  members 
of  this  equation  are  identically  the  same,  we  shall  obtain  the 
same  quotient  and  the  same  remainder  by  dividing  the  second 
member  by  x  —  x'.  But  U'  is  the  last  or  absolute  term  of  the 
transformed  equation.  It  follows,  therefore,  that  if  we  divide 
the  proposed  equation  by  x  —  x',  the  remainder  will  be  equal  to 
the  last  or  absolute  term  of  the  transformed  equation. 

Again,  let  the  quotient  arising  from  the  division  of  the  first 
member  of  (4)  by  a:  —  a;'  be  represented  by 
(a;  __  a:')"-' +  A' (a:  -  a;')"-2  _^  ....  R' (a:  —  a:') -f  T'. 
If  we  now  divide  this  quotient  by  a;  —  x',  every  term  will  be 
divisible  by  it,  except  the  last,  or  T',  the  coefficient  of  the  last 
term  but  one  of  the  transformed.  And,  since  the  remainder 
must  be  the  same  when  the  quotient  of  the  second  member  of 
(4)  by  a: — of  is  also  divided  by  a:— •«',  we  shall  obtain,  it  is 
evident,  the  coefficient  of  the  last  term  but  one  of  the  trans- 
formed equation  by  dividing  this  last  quotient  also  by  a;  —  x'. 
And  it  is  easy  to  see  that  thus,  by  successive  divisions,  all  the 
coefficients  of  the  transformed  equation  may  be  obtained. 

Thus,  resuming  the  equation  already  transformed,  viz. :  4  x^ — 
5a:^  +  7a; —  9  =  0,  and  dividing  by  a: -j-  1,  we  obtain  a  quo- 
tient 4  a;^  —  9  a;  -|-  16,  and  a  remainder  —  25.  Dividing  next 
this  quotient  by  a;  -(-  !»  we  obtain  a  new  quotient  4  a;  —  13,  and 


28S 


ELEMENTS   OF   ALGEBRA. 


a  remainder  29.  Dividing  next  this  last  quotient  by  a;  -(-  1,  the 
operation  terminates,  and  we  have  a  remainder  — 17.  The 
transformed  will  be,  therefore,  as  before, 

4^3  — 17^2  ^29  w  — 25  =  0. 

We  have,  then,  the  following  rule  by  which  to  obtain  the 
transformed  equation : 

Let  x'  be  the  quantity  by  which  the  roots  of  the  equation  are 
required  to  be  increased  or  diminished.  Divide  the  equation 
by  z  —  x\  or  x-\-x',  as  the  case  may  require,  and  the  quotient 
thus  obtained  by  the  same  quantity,  and  so  on  until  the  division 
terminates.  The  coefficient  of  the  first  term^  or  that  containing 
the  highest  poiver  of  the  transformed^  will  he  the  same  with  the 
coefficient  of  the  first  term  of  the  proposed  equation  ;  and  the 
successive  coefficients  will  be  the  remainders  arising  from  the 
successive  divisioTis  taken  in  a  reverse  order,  the  first  remainder 
being  the  last  or  absolute  term  of  the  transformed  equation. 

The  successive  divisions,  which  are  tedious  by  the  common 
method,  are  performed  in  a  very  concise  and  elegant  manner  by 
synthetic  division. 

Thus,  in  the  last  example. 


1 
1 

4—   5H 
—   4- 

f-  7—  9 
-9-16 

—  9-\ 

-  4- 

P  16  -  25 
-13 

—  13H 

—  4 

[-29 

—  17 

In  which  the  remainders,  as  before,  are  —  25,  29  and  —  17. 


Examples. 


1.  Transform  the  equation  b  x* -\- 28  a^ -j- 61  a^ -\- 22  x -- 1 
=0  into  another,  having  its  roots  greater  by  2  than  those  of 
the  proposed  equation. 


GENERAL  THEORY   OF   EQUATIONS. 


289 


1 

-2 


Operation. 

54-28  +  51  +  32-1 
__10  — 36  — 30  — 4 


5+18+15+  2 
—  10—16+   2 


5+   8  — 
-10  + 


1  + 
4 


4 


5—  2  + 
—  10 


12 

Ans.  b2^  —  123^  +  Sx'  +  4:X--b  =  0. 
2.  Find  an  equation  whose  roots  shall  be  less  by  ^  than  those 
of  the  equation  3x*  —  13a^~\-lx^  —  8x-—9  =  0, 

Placing  the  divisor  at  the  right,  as  in  ordinary  division,  and 
omitting  all  the  figures  in  the  first  column  except  the  first,  the 
calculations  may  be  more  conveniently  disposed,  thus : 
3    —13      +7      —8      —   9     I  1 
1       -  4  1       -   2^  U 

-11* 


-12 

1 

3 

-3f 

—  7 

-  f 

—  11 
1 

-3^ 

-n 

—  10 

1 

—  4 

12  o;  —  28  =  0  into  another, 


-    9 

Ans.  3  a:* 

3.  Transform  the  equation  a^ 
whose  roots  shall  be  less  by  4. 

In  the  use  of  synthetic  division  the  missing  term  in  this 
example  must  be  supplied  by  0  coefficient. 

Ans.  3^-{-12a^-{-S6x^l2  =  0. 

4.  Transform  the  equation  3a;*  —  4a;^  +  7a:^  +  8a:  —  12 
=  0  into  another,  whose  roots  shall  be  less  by  3. 

Ans.  3a;*  +  32a:3_|.  i33^^266a:  +  210=0. 
19 


290  ELEMENTS   OF   ALGEBRA. 

5.  Find  the  equation  whose  roots  shall  be  greater  by  2  than 
those  of  the  equation  x^ -\- 10  x^  +  A2  x^ -\- 86  x"^ -{- 10  x -j- 
12  =  0.  Ans.  x^^2x^  —  6x^—10x-\-S=0. 

6.  Find  the  equation  whose  roots  shall  be  less  by  ^  than  those 
of  the  equation  2x^^6x^-\-^x^  —  2x-\-l^=:0. 

Ans.  2x*  —  2a^  —  2x^  —  ^x  +  ^=:0, 

240.  The  transformation  above  is  the  only  one  for  which  we 
have  immediate  occasion.  There  are  others,  which  are  some- 
times useful,  which  we  will  here  explain. 

1.  From  what  has  been  done  we  may  readily  transform  an 
equation  into  another,  deprived  of  its  second  term. 

Resuming  the  general  equation  x""  -\-  K  a;"~^  -|-  ...  T  a:  -}- 
U  =  0,  and  putting  a;  =  w  -j-  :r',  developing  and  arranging  with 
reference  to  the  descending  powers  of  u^  we  have 

w''-(-(7ia;'  +  A)  w"-i =0. 

In  order  that  the  second  term  of  this  last  may  disappear,  its 
coefficient  must  be  equal  to  0.    We  shall  have  then  the  relation 

?za;'-|-A  =  0;  whence  a:' = . 

n 

Hence,  to  make  the  second  term  of  an  equation  disappear, 

P.  Divide   the  coefficient  of  the  second  term   by  the  highest 

power  of  the  unknown  quantity.     2°.   Transform  the  equation 

into  another  whose  roots  shall  be  less  or  greater  by  the  quotient 

thus  obtai7iedi  acco?'ding  as  the  sign  of  the  second  term  is  nega- 

tive  or  positive. 

Examples. 

Deprive  the  following  equations  of  their  second  terms : 

1.  s^—6x^4.4:X  —  l  =  0.       Ans.  a^  —  Sx  —  15==0. 

2.  a;*--8a:3+16a;2+7a;— 12  =  0. 

Ans.  x'  —  8x''  +  lx+18=:0, 

3.  a^^6x^-j-8x  —  2.  Ans.  a:^— 4a:  — 2  =  0. 

4.  ar'+15a;*+  12a;3_20a:2_|_  i42._25^  0. 

Ans.  a:^— 78a:«  +  412a:2_757a;  +  401  =  0. 


GENERAL  THEORY  OF  EQUATIONS.  291 

2.  An  equation  may  be  transformed  into  another,  the  roots 
of  which  shall  be  any  multiple  of  those  of  the  proposed  equa- 
tion. 

In  the  general  equation  a;"  -}-  A  a:""^  -f-  .  .  .  .  T  a;  -f-  U  =  0, 

X  X 

let  a:  =  — .    Substituting  -  for  a;,  and  clearing  of  fractions,  we 

obtain 

ar-'^- A77ta;'^i  +  Bm2af-2 T mP-'^ x -{-V 7nr  =  0, 

the  roots  of  which  are  m  times  greater  than  those  of  the  pro- 
posed. The  required  equation  is,  therefore,  found  by  multiply- 
ing the  second  coefficient  of  the  proposed  by  w,  the  third  by 
m^,  and  so  on,  ?»  Representing  the  number  of  times  the  required 
roots  are  to  exceed  those  of  the  proposed. 

Ex.    Find   an   equation  whose   roots   shall   be    three   times 
larger  than  those  of  the  equation  3?  —  Q  x'^  -{-Sx —  9  =  0. 
Ans.  a:^— 18a:2  +  72a:  — 243  =  0. 

By  means  of  this  principle  we  may  transform  an  equation 
with  fractional  coefficients  into  another,  whose  coefficients  shall 
be  integral.  In  order  to  this,  we  have  merely  to  transform  the 
proposed  equation  into  another,  the  roots  of  which  shall  be 
equal  to  those  of  the  proposed  multiplied  by  the  least  common 
multiple  of  the  denominators  of  the  fractions. 

Ex.  1.  Transform  the  equation  a:^-}-K^^"— t^  +  2  =  0  in- 
to another,  the  coefficients  of  which  shall  be  integral. 

Ans.  a:»-|-4a:2  —  36a;-f-3456  =  0. 

In  this  example  the  number  6,  when  raised  to  the  square,  will, 

it  is  evident,  be  divisible  by  4.     The  fractional  coefficients  may, 

therefore,  be  removed,  and  a  more  simple  result  obtained,  if 

we  multiply  by  the  successive  powers  of  6.     It  will  be  easy  to 

apply  the  like  simplification  to  other  cases. 

7  11         25 

Ex.  2.  Transform  the  equation  j?  —  -x'^  -\-  ^-x  —  —  =0, 

into  another,  the  coefficients  of  which  shall  be  integral. 

Ans.  a:^  — .Ua;2-|-lla;  — 75  =  0. 


292 


ELEMENTS    OF    ALGEBRA. 


3.  In  like  manner  an  equation  may  be  transformed  into 
another,  the  roots  of  which  shall  be  the  reciprocals  of  the  pro- 
posed equation. 

In  order  to  this,  substituting  -  for  x  in  the  general  equation, 

clearing  of  fractions,  and  reversing  the  order  of  the  terms,  it 
becomes 

U  af'  4-  T  a:"-!  4-  S  a;"-2  ^ A  a:  4-  1  ==  0, 

the  roots  of  which  are  reciprocals  of  those  of  the  proposed  equa- 
tion. 

The  transformation  is  therefore  effected  by  simply  changing 
the  order  of  the  coefficients  of  the  proposed  equ&tion. 

If  the  coefficients  of  the  proposed  equation  are  the  same, 
whether  taken  in  the  reverse  or  direct  order,  the  transformed 
equation,  it  is  evident,  will  be  identical  with  tlie  given  equation, 
and  will  furnish  the  same  series  of  roots. 

Equations  of  this  description  are  called  recurring  equations, 
or,  from  the  form  of  the  roots,  reciprocal  equations. 

4.  From  what  has  been  done,  an  equation,  it  is  evident,  may 
be  transformed  into  another,  whose  roots  shall  be  greater  or 
less  than  the  reciprocals  of  the  proposed  equation.  In  order  to 
this,  we  reverse  the  order  of  the  coefficients,  and  then  apply  the 
process  of  art.  237. 

Ex.  1.  Transform  the  equation  oi?  —  1  x  -\-  7  into  another, 
the  roots  of  which  shall  be  less  by  1  than  the  reciprocals  of 
those  of  the  proposed.        Ans.  1  a?  -{-l^a?  -\-l  x  -\-\  =  0. 

Ex.  2.  Find  the  equation  whose  roots  shall  be  the  reciprocals 
of  those  of  3  2:*—  13  2:5  +  7 2:^  —  8 2;  —  9  =  0,  increased  by  2. 
Ans.  — 92:^  +  642:3  __i6ia.2^151^_23  =  0. 

LDIITS   OF   THE    ROOTS. 

241.  The  methods  for  resolving  numerical  equations  of  any 
degree  consist,  in  general,  in  substituting  particular  numbers 
for  X  in  the  equations,  in  order  to  see  if  these  numbers  will 
verify  them ;  or  for  the  purpose  of  determining  the  initial  figures 


GENERAL  THEORY  OF  EQUATIONS.  293 

of  the  roots.  It  is  important,  therefore,  as  a  preliminary  step, 
to  determine  the  limits  between  which  the  roots  of  an  equation 
are  to  be  sought. 

1.  A  number  numerically  greater  than  the  greatest  positive 
root  of  an  equation  is  called  a  Superior  Limit  to  the  positive 
roots.  A  number  numerically  greater  than  the  greatest  nega- 
tive root,  abstraction  being  made  of  the  sign,  is  called  a 
Superior  Limit  to  the  negative  roots. 

The  extreme  limits  to  the  positive  roots,  it  is  evident,  will 
be  0  and  +  oo  ,  and  those  of  the  negative  roots  0  and  —  oo  ,  the 
sign  OD,  being  used  to  denote  infinity.  In  practice,  much  nar- 
rower limits  than  these  will  be  required. 

Since  the  lai^est  of  the  positive  roots  when  substituted  for  ar, 
reduces  the  equation  to  0,  it  follows  that  a  number  greater  than 
this,  when  substituted  in  like  manner  for  x,  will  give  a  positive 
result.  If,  therefore,  a  number  substituted  for  x  in  an  equation 
gives  a  positive  result ^  then  this  number  will  be  a  superior  limit 
to  the  positive  roots. 

1.  It  may  be  shown  that  in  any  equation  the  greatest  Tiega- 
tive  coefficient  ii^reaseA  by  unity  is  a  superior  limit  to  the  posi- 
tive roots.     A  much  nearer  limit  may,  however,  be  found. 

2.  Let.  us  take  the  general  equation, 

a:"  4-  il  a:"-i  -f  .  .  .  —  D 2;"-'^ ™  Pa;"-*  .  .  .  .  U  =  0, 

and  let  —  D  a;"-*"  be  the  first  negative  term,  and  — '  P  the  great- 
est negative  coefficient;  then,  if  all  the  terms  after  the  first 
negative  term  are  negative,  the  sum  of  these  negative  terms, 
it  is  evident,  must  be  equal  to  the  preceding  positive  terms. 
And  any  value,  which,  substituted  for  x,  will  make  the  sum  of 
the  positive  terms  greater  than  the  sum  of  the  negative  terms, 
will  be  a  superior  limit.  And,  P  being  the  greatest  negative 
coefficient,  for  a  still  stronger  reason,  any  number  will  be  a 
superior  limit,  which  substituted  for  x  gives 

^n^p(^n_.^^.-r-l^ OT  +    1),         (1) 

or,  since  the  right  hand  factor  of  this  inequality  is  a  progression 
by  quotient,  art.  176, 

25* 


294  ELEMENTS    OF   ALGEBRA. 

But  this  inequality  will  be  satisfied  if  we  have 

->KS) 

or,  reducing         (^  —  1)  a;'-^^  P;  (3) 

but  X  —  1  is  less  than  2:,  and  by  consequence  {x —  1)  {x  —  1)''"^ 
or,  (a; — l)'"is  less  than  {x  —  l)a;'-^;  the  inequality  (3)  will, 
therefore,  be  satisfied  if  we  have 

2 
{x-^\Y  =,  or  >  P,  or  a;  =,  or  >  ?'•+  1. 

Thus,  to  obtain  a  superior  limit  of  the  positive  roots,  we  in- 
crease by  unity  the  root  of  the  greatest  negative  coefficiejit,  whose 
index  is  the  number  of  terms  which  precede  the  first  negative 
term. 

EXAMPLES. 

Find  superior  limits  to  the  roots  of  the  following  equations : 

1.  ic^  +  7a:^—12a;3  — 49^:24-520;  — 13  =  0. 

i-  1 

Ans.  P'-.f  1  =  (49)2+1=8. 

2.  a:*  — 5a;3  +  37a;2_3x -1-39  =  0.  Ans.  6. 

3.  3  a;3  —  2  a:2  —  1 1  a;  +  4  =  0.  Divide  the  equation  first 
by  3,  then  applying  the  rule  the  limit  will  be  5. 

4.  x^-\-\\x'^  —  25  a;  —  67  =  0.  The  greatest  negative  co- 
efficient, is  that  of  tP^  or  67,  and  the  missing  term  being  counted, 

j_  \_ 

which  is  necessary,    r  =  3 ;  whence  P*"-]-  1  =  (67)  ^  -)-  1  =  6- 

5.  x^  +  11  0:2  -  25  a:—  61  =  0.  Ans.  5. 
242.  A  number  numerically  less  than  the  least  positive  root 

of  an  equation  is  called  an  inferior  limit  to  thd  positive  roots. 

In  the  preceding  examples  we  have  regarded  0  as  the  inferior 
limit  of  the  positive  roots.  Thus  the  positive  roots  of  the  first 
example  are  all  comprised  between  0  and  8.  A  nearer  inferior 
limit  may,  however,  be  found. 

Let  X  =  0  be  any  equation,  and  in  this  equation  let  us  put 


GENERAL  THEORY  OF  EQUATIONS.  295 

x  =  -.    We  shall  obtain,  it  is  evident,  a  derived  equation  Y  =  0 

y 

in  which  the  greatest  value  of  y  will  correspond,  to  the  least 
value  of  X.    If  then  we  find  the  superior  limit  L  to  the  roots  of 

the  equation  Y  =  0,  the  reciprocal  of  this,  ory-,  will  be  the  in- 
ferior limit  to  the  roots  of  the  proposed  equation,  or  X  =  0. 

Thus,  let  it  be  proposed  to  find  the  inferior  limit  of  the  posi- 
tive roots  of  the  equation  7?  —  42  x"^ -{- 4:^1  x  —  49  =  0. 

Putting  a:  =  - ,  we  have,  for  the  transformed  equation, 

which  gives,  by  the  rule,  10  for  the  superior  limit  of  its  posi- 
tive roots.     Thus  the  inferior  limit  to  the  positive  roots  of  the 

proposed,  will  be  j^. 

243.  The  particular  form  of  the  equation  may  sometimes 
suggest  artifices,  by  means  of  which  closer  limits  may  be  ob- 
tained than  those  given  by  the  preceding  rules. 

Thus,  the  equation  re*  -f  11  a:^  —  25  a;  —  61  =  0,  may  be  put 
under  the  form 

in  which  it  is  evident  that  a:  =  3,  or  any  number  greater  than 
3,  will  give  a  positive  result.  We  shall  have  3,  therefore,  for 
the  superior  limit  to  the  positive  roots,  which  is  much  nearer 
than  5,  the  limit  obtained  by  the  rule. 

The  equation  x^-^ba^  +  Sl  x^ —  S  x  +  Sd  =  0,  maybe  put 
under  the  form 

7^{x-5)  +  31x(^x-~^+29  =  0; 

and  the  equation  2^+lx'-l22^  —  49a^-\-b2x^l3  =  0, 
under  the  form 


296  ELEMENTS   OF   ALGEBRA. 

It  is  evident  that  5,  and  any  number  greater  than  5,  sub- 
otituted  for  x  in  the  first  of  these  equations,  and  4,  or  any 
number  greater  than  4,  substituted  for  x  in  the  second,  will  give 
positive  results.  We  have,  therefore,  5  and  4  respectively  for 
the  limits,  instead  of  6  and  8  obtained  by  the  rule. 

The  artifice  consists  in  decomposing  the  equation  into  parts, 
each  of  which  is  composed  of  two  factors,  the  first  a  positive 
monomial,  and  the  other  a  binomial  in  a;,  the  second  term  of 
which  is  negative,  and  then  determining  x  in  such  a  manner, 
that  all  the  factors  within  parentheses  shall  be  positive. 

244.  It  remains  to  find  the  superior  and  inferior  limits  to  the 
negative  roots.  In  order  to  this,  we  transform  the  proposed 
equation  into  another  whose  roots  shall  be  the  same  as  the  pro- 
posed with  contrary  signs.  The  limits  of  the  positive  roots  of 
this  equation,  taken  witn  a  contrary  sign,  will  be  the  limits  to 
the  negative  roots  of  the  proposed  equation. 

Thus,  let  it  be  proposed  to  find  the  limits  to  the  negative  root 
of  the  equation  :j^  —  7a;-[-7  =  0.  Putting  x-=.  —  x^  or,  which 
is  the  same  thing,  changing  the  signs  of  the  alternate  terms, 
the  missing  term  being  supplied,  we  have 

the  limits  to  which  are  4  and  3  respectively.     The  negative 
root  of  the  proposed  will,  therefore,  lie  between  —  4  and  —  3. 

245.  In  the  preceding  numbers  we  have  found  the  limits 
between  which  the  roots  cHl  lie.  When  the  roots  of  the  pro- 
posed are  incommensurable  we  shall  wish  to  find  limits  between 
which  the  iiidividual  roots  are  situated,  in  order  to  determme 
more  readily  the  initial  figures  of  the  roots. 

Let  a,  b,  c,  &c.,  be  the  real  roots  of  an  equation  in  the  order 
of  their  magnitude,  so  that  we  have  a^b,  b^  c,  &c. ;  and 
let  a',  b\  c',  be  a  series  of  numbers,  such  that  a!  is  greater  than 
a,  b'  2l  number  comprised  between  a  and  b,  so  that  we  have 
b'  <^a,  b'"^  b,  and  so  on. 

The  original  equation,  it  is  evident,  will  be 

{x  —  a){x  —  b){x  —  c) =0. 


GENERAL  THEORY  OF  EQUATIONS.  297 

If  now,  in  this  equation,  we  substitute  a'  for  a;,  it  becomes 
(a'  — a)  (a' —  i)  (a'-r-c)  .    .    .   .=0; 

and  since  o!  is  greater  than  a,  3,  c,  &c.,  the  factors  will  each  be 
positive,  and  hence  their  product  will  be  'positive. 

Again,  let  V  be  substituted  for  2;,  the  equation  becomes 
^}/^a){b'  —  b)  {b'  —  c) =0; 

here,  since,  b'  is  less  than  a,  but  greater  than  i,  c,  &c.,  the  first 
factor  will  be  negative  and  the  rest  positive  ;  the  product,  there- 
fore, will  be  negative. 

If,  again,  c'  be  substituted  for  x,  the  equation  becomes 

{c'—a){c'  —  b){c'-'c)  [   .   .   .   .  =0; 

here,  since  c'  is  less  than  a  and  b,  but  greater  than  c,  &c.,  the 
first  two  factors  will  be  negative  and  the  rest  positive;  the 
product  will,  therefore,  be  positive. 

Hence,  1°,  If  a  quantity,  greater  than  t/ie  greatest  real  root, 
be  substituted  for  x,  the  result  will  be  positive. 

2°.  If  quantities  intermediate  between  the  roots,  beghining 
with  the  greatest,  be  substituted,  the  results  will  be  alternately 
negative  and  positive. 

Ex.  1.  The  roots  of  the  equation  a^  —  13  x  -|-  12  =  0,  are 
3,  1,  and  —  4.  Substitute  4,  2,  0,  and  —  5  for  x,  and  observe 
the  signs  of  the  results. 

Ex.  2.  Make  a  like  substitution  in  the  equation  a^  —  da^  -{- 
2  a;  +  8  =  0,  the  roots  of  which  are  4,  *2,  and  —  1. 

From  the  preceding  principles  it  follows, 

1°.  If  two  numbers  be  successively  substituted  for  x  in  any 
equation,  and  give  results  with  different  signs,  then  between 
these  numbers  there  must  be  otic,  three.  Jive ,  or  some  odd  number 
of  roots. 

2°.  But  if  the  numbers  substituted  for  x  give  the  same  sign, 
then  between  these  numbers  there  will  be  either  no  root,  or 
there  will  be  two,  four,  or  some  even  number  of  roots. 

3®.  If  any  quantity  q  and  every  quantity  greater   than  q 


298  ELEMENTS    OF   ALGEBRA. 

renders  the  result  positive,  then  q  is  greater  than  the  greatest 
root  of  the  equation,  and  will  be  a  superior  limit  of  the  roots. 

4°.  Hence,  if  the  signs  of  the  alternate  terms  are  changed, 
and  if  p  and  every  quantity  greater  than  p  renders  the  result 
positive,  then  — ^  is  less  than  the  least  root,  and  will  be  an 
inferior  limit. 

5°.  If  the  degree  of  the  equation  be  even,  the  substitution  of 
a  number  less  than  the  least  root  will  give  a  positive  result. 
But  if  the  degree  be  odd,  the  result  will  be  negative. 

246.  The  substitution  of  the  natural  series,  0,  1,  2,  3,  &c., 
taken  negatively  as  well  as  positively,  will  enable  us  to  discover 
the  position,  and  determine,  in  general,  the  initial  figure  of 
the  real  roots. 

Ex.  1.  Let  it  be  required  to  find  one  of  the  roots  of  the 
equation  oi?  —  ^x^  —  Qx-\-Q  —  0. 

Substituting  0  for  x,  we  have  8  fpr  the  result.  Substituting 
next  1,  the  result  will  be  —  1.  There  will  be  a  root,  therefore, 
between  0  and  1,  and  very  near  1.  Try  next  .9.  Putting  .9 
for  X,  the  result  is  .089.  There  is,  therefore,  a  root  between 
.9  and  1.  We  shall  have  then  .9  for  the  initial  figure  of  the 
root. 

Ex.  2.  Find  the  first  figure  of  one  of  the  roots  of  the  equation 
a;4-|- 3 a;'^  +  2a:2 -f-6  a;  —  143  ^  q. 

Ans.  The  substitution  of  2  gives  a  negative,  and  of  3  a  posi- 
tive result.  There  is,  therefore,  a  root  between  2  and  3. 
Hence  2  is  the  first  figure  of  the  root. 

Ex.  3.  Find  the  first  figure  of  one  of  the  roots  of  the  equation 
x'  +  x'J^x-im^O,         i  Ans  4. 

Ex.  4.  Find  the  first  figure  of  one  of  the  roots  of  the  equation 
a:3_)_  1,53.2^  .32:  — 46=0.  Ans.  3. 

Ex.  5.  Find  the  first  figure  of  one  of  the  roots  of  the  equation 
a:^  — 12a:  +  8  =  0.  Ans.  .6. 


GENERAL  THEORY   OF   EQUATIONS.  299 


LIMITING    EQUATION.       EQUAL   ROOTS. 

247.  An  equation,  the  roots  of  which  are  intermediate  be- 
tween those  of  a  proposed  equation,  will  have  for  its  roots  limits 
to  those  of  the  proposed  equation,  and  is,  therefore,  called  the 
separating  or  limiting  equation. 

Let  a,  h,  c,  &c.,  taken  in  the  order  of  their  magnitude,  be  the 
roots  of  the  -equation 

a-n  _  A  3,n_i  _|_  B  ^n-2  ^ T  2:  +  U  =  0 ;      (1) 

it  is  required  to  find  an  equation  the  roots  of  which  shall  lie 
between  or  separate  those  of  the  proposed  equation. 

Diminishing  the  roots  of  the  proposed  by  x\  we  put  x  =  u 
-j-  x',  and  developing,  as  in  art.  237,  it  becomes 

w"+A'm"-i+ T'2^-f-U'=0, 

in  which  the  coefficient  of  the  last  term,  U',  will  be 

x'-  -f  A  a:'"-!  +  P  x'"-^  +  .   .   .  T  aT  +  U ; 
and  the  coefficient  T'  of  the  term  before  the  last  will  be  . 
7^a:"-l-|-(7^  — 1)  Aa;"'-2  +  (7i  — 2)  Ba;'"-3+  .    .    .    .  T.    (2) 

But  a,  b,  c,  &c.,  being  roots  of  equation  (1),  the  roots  of  the 
transformed  will*  be  {a  —  xf),  {b  —  x'),  {c  —  x'),  &cc.  Hence, 
by  art.  229,  the  coefficient  of  the  last  term  but  one  of  the  trans- 
formed will  be  the  sum  of  the  products  of  every  n  —  1  of 
these  roots  with  their  signs  changed.  Thus  this  coefficient 
will  be 

{x'  —  b)  {ctf  —  c)  {x'  —  d)  to  n—1  factors,  "j 

--{:xf —a){x' —  c){x'  —d)       "  "  | 

-^{3f  -a)  {xf  —  b)  {x'  —  d)       "  "  I        (3) 

--l3f  —  a){x'  —  b){x'—c)       "  « 

--&C.  J 

all  the  factors  save  one  occurring,  of  course,  in  each  term. 

But  the  expressions  (2)  and  (3)  are  equal,  since  they  are  but 
different  expressions  for  the  same  thing,  viz.,  the  last  coefficient 
but  one  of  the  transformed  equation.  Hence,  whatever  changes 
are  produced  by  substitution  in  one  of  these  expressions,  the 


. . .  =  + 

•  + 

•  += 

+ 

.   .   .  =  — 

+ 

+  = 

I  — 

.    .    .    =  — 

.  — 

+= 

=+ 

300  ELEMENTS   OF   ALGEBRA. 

same  changes  will  be  produced  by  a  like  substitution  in  the 
other. 

If  we  now  substitute  a  for  x'  in  each  term  of  (3),  all  the 
terms,  except  the  first,  will  vanish,  and  the  coefficient  will  be 
reduced  to  {a  —  h)  [a  —  c)  {a  —  d)  .  .  .  in  which  the  signs  of 
the  factors  are  each  positive,  since  a  is  greater  than  b,  c,  d,  &c. 
Substituting  b,  in  like  manner,  all  the  terms  vanish  except  the 
second,  in  which  the  sign  of  the  first  factor  will  be  Jiegative,  and 
those  of  the  rest  positive.  For  the  substitution  of  a,  b,  c,  &c., 
successively,  we  shall  have  the  following  results. 
1st,  a,  {a  —  b)  {a  —  c)  {a  —  d) 
2d,b,  {b  —  a)  {b  —  c)  {b  —  d) 
3d,  c,  {c  —  a)  {c  —  b)  {c  —  d) 
^th,d,  (d  —  a)    {d  —  b)    {d--c) 

Sec.,  &c. 
and  by  consequence  the  same  changes  of  sign  will  result  from 
a  like  substitution  in  (2). 

If  then  we  put  (2)  =  0,  and  write  x  for  x',  it  becomes 

wx"-i+(?z  — 1)  Aa:"-2  +  B(?i  — 2)a:"-3+ T  =  0.  (4) 

-And  since,  from  what  has  been  done,  the  substitution  of  a,  b,  c, 
&c.,  in  this  last  gives  results  alternately  positive  and  negative, 
its  roots  will  be  intermediate  between  these  quantities,  that  is, 
between  the  roots  of  equation  (1). 

Equation  (4)  will,  therefore,  be  the  limiting  equation  to  equa- 
tion (1) ;  and  since  the  former  is  the  derivative  of  the  latter, 
in  general,  the  derivative  of  an  equation  will  be  its  limiting 
equation. 

Ex.  1.  What  is  the  limiting  equation  to  a;^  —  1  x^  -\-bx^  -\- 
31a:-30  =  0?  Ans.  4a:3  —  21  a:2+ 10a:+ 31  =  0. 

248.  The  roots  of  equation  (1)  in  the  preceding  article,  being 

represented  by  a,  b,  c,  &c.,  respectively,  if  b',  c',  d',  &c.,  in  like 

manner  represent  the  roots  of  equation  (4),  the  roots  of  the  two 

equations  arranged  in  the  order  of  their  magnitude  will  stand 

thus : 

a,  b',  b,  cfj  c,  d',  &c. 


GENEEAL  THEORY  OF  EQUATIONS.  301 

Here,  if  the  difference  between  a.  and  b  is  zero,  the  difference 
between  a  and  b'  must  also  be  zero ;  that  is,  if  a  is  equal  to  i,  it 
must  also  be  equal  to  b' ;  hence  the  factor  x  —  a  will  be  found 
both  in  the  proposed  and  its  limiting  equation.  The  two  equa- 
tions will,  therefore,  have  a  common  measure  x  —  a. 

Again,  if  b  and  c  are  each  equal  to  a,  then  b',  c',  will  also  be 
each  equal  to  a,  and  the  two  equations  will  have  a  common 
measure  {x  —  of,  and  so  on. 

Conversely,  if  a  proposed  equation  and  its  derivative  have  a 
common  measure  x  —  «,  the  proposed  will  have  two  roots,  each 
equal  to  a.  If  they  have  a  common  measure,  {x  —  af,  the 
proposed  will  have  three  roots,  each  equal  to  a,  and  so  on. 

To  determine,  therefore,  whether  an  equation  has  equal  roots, 
we  form  the  derivative  or  limiting  equation^  and  then  seek  the 
greatest  commxm  divisor  of  the  two  equations. 

The  factors  of  this  common  divisor  being  determined,  it  is 
evident  they  must  enter  each  once  more  into  the  proposed,  and 
thus  the  number  and  value  of  the  equal  roots  will  be  deter- 
mined. 

Suppose  the  greatest  common  divisor  of  the  proposed  and  its 
derivative  to  be  {x  —  af  {x  —  bf  {x  —  c).  The  proposed  will 
have  4  roots  equal  to  a,  3  equal  to  b,  and  2  equal  to  c. 

Ex.  1.  Find  the  equal  roots  of  the  equation  3  a:*  —  10  a:' 
+  15  a;  +  8  =  0. 

The  derivative  is  15  a:*  —  30  a:^  -j-  15  =  0.  And  the  greatest 
common  divisor  is  x^  -\- 2  x -{- \^  ox^  {x '\-  \f.  The  proposed, 
therefore,  has  2  roots  equal  each  to  —  1. 

Ex.  2.  Find  the  equal  roots  of  the  equation  x*  —  14  a:^  -|-61 
a:2_84a;  +  36  =  0. 

The  greatest  common  divisor  between  the  proposed  and  its 
derivative  is  ar'  —  7  a;  +  6  =  (a;  —  6)  {x  —  1). 

Ans.  Two  roots  equal  to  6  and  two  equal  to  1. 

Ex.  3.  Find  the  equal  roots  of  the  equation  a:*  —  2  a:*  -|-  1 
=  0.  Ans.  Two  equal  to  1,  and  two  equal  to  —  1. 

26  , 


302  ELEMENTS    OF   ALGEBRA. 

Ex.  4.  Find  the  equal  roots  of  the  equation  a:^  —  2x^  —  4  a: 
+  8  =  0.  Ans.  Two  equal  to  2. 

Ex.  5.  Find  the  equal  roots  of  the  equation  a:*  —  6  a:'  +  8 
x^  -}-  6x  —  9  =  0.  Ans.  Two  equal  to  3. 

IMAGINARY  AND  REAL  ROOTS.      STURM's   THEOREM. 

249.  In  the  search  for  the  roots  of  an  equation,  it.  is  of  the 
first  importance  to  determine,  at  the  outset,  the  precise  number 
of  real  roots  the  proposed  equation  contains,  m  order  that  the 
substitutions  required  in  the  solution  may  be  restricted  within 
the  narrowest  possible  limits.  To  separate  the  real  and  im- 
aginary roots  of  a  proposed  equation  so  as  to  determine  the 
exact  number  of  each,  is  a  problem  of  great  and  acknowledged 
difficulty.  It  entirely  baffled  the  skill  of  mathematicians,  until 
in  1829  it  was  completely  solved  by  the  beautiful  Theorem  of 
Sturm,  which  w^e  shall  now  explain. 

LetV  =  a:"  +  Aa:"-i  +  Ba:"-2+  .  .  .  Ta:  +  U  =  0, 
be  an  equation  of  the  nth.  degree  with  no  equal  roots,  and  Vj  =  0 
be  its  derivative  or  limiting  equation. 

We  now  apply  to  V  and  V^  the  process  for  finding  their  greatest 
common  divisor  until  a  remainder  is  obtained  independent  of  a?, 
observing,  however,  to  change  the  signs  of  each  remainder  as 
we  proceed. 

Let  the  series  of  remainders,  with  their  signs  changed,  be 
represented  by  Vg,  Vg,  V4  .  .  .  V„,  V„  being  the  last  remainder, 
or  that  which  is  independent  of  x. 

Let  p  and  q  be  any  numbers  taken  at  pleasure,  of  which  p  is 
the  greater.  Let  p  be  substituted  in  the  place  of  x  in  the  func- 
tions V,  Vi,  V2 V„,  and  write  in  order  in  one  line 

the  signs  of  the  results.  Substitute  next  q  in  the  same  manner, 
and  write  in  order  the  signs  of  the  results.  The  difference  in 
the  number  of  variations  between  the  two  rows  of  signs  resulting 
from  these  substitutions,  will  be  equal  to  the  number  of  real  roots 
of  the  equation  V=;=  0,  comprised  between  the  nurribers  p  and  q. 


GENERAL  THEORY  OF  EQUATIONS.  303 

This  is  the  Theorem  of  Sturm,  which  we  now  proceed  to 
demonstrate. 

In  order  to  this,  let  Q,  Qi,  Q2,  &c.,  represent  the  quotients  ob- 
tained in  the  successive  divisions  of  V  by  Vj,  Vj  by  Vg,  and  so 
on.     From  the  manner  in  which  these  functions  are  derived, 
we   shall  have  this  series  of  equations : 
V=Q  V,-V/ 

*  n-2  =  Qn-2   *  n-1 »  n 

This  being  premised,  we  remark  • 

1°.  No  two  coTisecutive  functions  can  become  0,  or  vanish^  far 
the  same  value  ofx. 

For  if  two  consecutive  functions  Vg,  Vg,  for  example,  can  at 
the  same  time  become  equal  to  0,  then  we  shall  have  ¥4  =  0; 
and  Vg,  V4  being  each  equal  to  0,  then  V5  will  be  equal  to  0, 
and  so  on,  until  finally  V„  or  the  last  remainder  will  be  equal  to 
0,  which  is  impossible,  since  this  remainder  is  independent  of  2:, 
and  cannot  be  affected  by  any  change  in  the  value  of  x. 

2°.  If  one  of  the  functions,  Vg,  for  example,  becomes  0  for  a 
particular  value  of  x,  then  the  adjacent  functions  between  which 
it  is  placed,  have  for  that  value  contrary  signs. 

For  we  have  V2==Q2  V3  — V^; 
hence,  if  V3  =  0,  Vg  =  —  V4. 

Thus  the  adjacent  functions  have  in  this  case  contrary  signs. 

Let  now  p  be  greater  than  the  greatest  root,  and  negative 
roots  being  regarded  as  less  than  corresponding  positive  ones, 
let  q  be  less  than  the  least  root  of  the  equations 

V  =  0,Vi  =  0,V2  =  0 v_2  =  o, 

and  let  q  increasing  by  insensible  degrees  until  it  reaches  the 
value  of  p,  be  substituted  successively  for  x  in  these  equations. 
We  remark  again, 

1°.  So  long  as  q  remains  less  than  the  least  root  of  the  equa- 


304  ELEMENTS    OF   ALGEBRA. 

tions,  no  change  will  be  produced  in  the  signs  by  its  substitu- 
tion. 

2°.  But  when  q  in  the  process  of  its  increase  becomes  equal 
to  the  least  root  of  the  equations,  in  Vg,  for  example,  then  for 
this  value  V3  vanishes.  But  there  will  be  no  change  in  the 
number  of  variations  of  the  signs  produced  by  this  circumstance. 

Indeed,  q  being  still  less  than  the  least  root  of  Vg  and  V4,  the 
signs  of  these  will  not  change  by  the  substitution  which  causes 
Vg  to  vanish.  And  since  when  Vg  vanishes,  Vg  and  V4  have 
necessarily  opposite  signs,  the  signs  of  the  three  consecutive 
functions  must  before  have  been,  either 

+  =F  -  1   "    }  -  ±  + 
which  give  each,  whichever  of  the  double  signs  is  the  true  one, 
one  permanence  and  one  variation. 

Now,  when  Vg  vanishes,  the  signs  become 

V2,  V 3,   V 4  )  (   V2,  Vg,  V4 

+  0  - 1   "    I  -  0   + 

in  each  of  which  there  is  still  one  variation. 

If  q  now  becomes  greater  than  the  least  root  of  Vg,  but  is  still 
less  than  the  roots  of  Vg,  V4,  the  signs  of  Vg,  V4  will  remain  as 
they  were  before  ;  but  the  sign  of  Vg  will  change.  The  three 
consecutive  signs  will  then  be 

+  db— ,or  — =F+, 
still  exhibiting^one  permanence  and  one  variation. 

The  same  reasoning  obviously  applies  to  any  of  the  inter- 
mediate functions  between  V  and  V„. 

The  function  V„,  being  the  last  remainder  and  independent 
of  a;,  can  undergo  no  change  of  sign,  whatever  value  is  sub- 
stituted for  X.  It  follows,  therefore,  that  there  can  he  no  change 
in  the  number  of  variations  of  the  signs,  unless  it  arise  from  a 
change  of  sign  in  the  primitive  function. 

2r°.  Of  the  two  equations  V  =  0,  Vi  =  0,  one  will  necessarily 
be  of  an  even  degree  and  the  other  odd.  If,  therefore,  a  number 
less  than  their  least  roots  be  substituted  in  them,  the  results 


GENERAL  THEORY  OF  EQUATIONS.  305 

(art.  245)  will  be  of  different  signs.  Let  us  then  suppose  next 
that  the  value  of  q  in  its  increase  has  now  become  greater  than 
the  least  root  of  V  =  0,  but  is  still  less  than  the  least  root  of 
Vi  =  0,  the  roots  of  the  latter  (art.  247)  being  necessarily- 
greater  than  the  least  root  of  the  former.  When  q  passes 
the  least  root  of  V  =  0,  there  will  be  a  change  of  sign  in  this 
equation ;  thus  the  results  of  the  substitution  in  the  equations 
V  =i  0,  Vi  t=  0  will  now  have  the  same  sign.  That  is,  the 
results  of  the  substitution  in  these  two  equations,  which,  before 
exhibited  a  variation,  will,  in  its  stead,  exhibit  a  permanence. 
And  by  consequence  the  whole  number  of  variations  is  in  this 
case  diminished  by  unity. 

If  q  goes  on  to  increase  until  it  has  passed  the  least  root  of 
Vi  =  0,  this  function  will  change  sign,  so  that  V  and  Vi  will 
give  different  signs,  but  will  again  have  the  same  sign  when  the 
second  root  of  V  =  0  is  passed.  Thus  there  will  be  no  change 
in  the  number  of  variations  until  the  second  root  of  V  =  0  is 
passed,  when  the  number  of  variations  will  again  be  diminished 
by  unity. 

In  like  manner  it  may  be  shown  that  when  q  in  its  progress 
towards  p  passes  successively  each  of  the  remaining  roots  of  the 
primitive  function  V  =  0,  the  number  of  variations  in  each  case 
will  be  diminished  by  unity. 

It  follows,  therefore,  since  no  change  in  the  number  of  varia- 
tions is  made  when  any  of  the  functions  except  the  primitive 
are  reduced  to  zero,  that  the  difference  in  the  number  of  varia- 
tions,  when  p  and  q  are  successively  substituted  in  the  functions 
V,  Vj,  Vg,  (f*c.,  will  always  be  equal  to  the  number  of  real  roots 
comprised  between  p  and  q. 

This  is  the  proposition  which  was  to  be  demonstrated.  We 
will  illustrate  by  an  example. 

For  this  purpose  let  us  form  the  equation  whose  roots  shall  be 
1,  2,  3,  and  4.     It  will  be  a:*  —  10  3:^  +  35 a;^ -  50  a;  +  24  =0. 

The  functions  formed  according  to  the  rule,  and  their  roots, 

when  the  functions  are  made  equal  to  0,  will  be  as  follows : 
20 


306  ELEMENTS    OF   ALGEBRA. 

Functions.                   '  Roots. 

V  :=    2;4_10a:«-f-35a:'^  — 50a;  +  24  1,     2,    3,     4 

Vi  =  4  a:^  —  30  a:^  +  70  a:  —  50  1.3,  2.5, 3.6  nearly 

V2  =  5a;2  — 25a;  +29  1.8,3.1 

V8  =  2a:  -     5  2.5 
V,  =  9 

Since  the  least  root  of  the  functions  is  1,  we  will  begin  the 

substitutions  with  a  number  less  than  this  .8,  for  example. 


V     Vi 

V, 

Va 

V4 

=  .8  i 

gives  -\-    — 

+ 

— 

+ 

4  variations. 

.9 

(( 

+  - 

+ 

— 

+ 

4 

(( 

1 

<( 

0    — 

+ 

— 

+ 

1.1 

(( 

—  — 

+ 

— 

+ 

3 

(( 

1.9 

(( 

-  + 

— 

— 

+ 

3 

(( 

2 

(( 

0    + 

— 

— 

+ 

2.1 

(( 

+  + 

— 

— 

+ 

2 

(( 

2.5 

(( 

+    0 

— 

0 

+ 

2 

it 

2.9 

(( 

+  - 

— 

+ 

+ 

2 

« 

3 

(( 

0    — 

— 

+ 

+ 

3.1 

(( 

—  — 

— 

+ 

+ 

1  variation. 

3.9 

(( 

-  + 

+ 

+ 

+ 

1 

(( 

4 

(( 

0    + 

+ 

+ 

+ 

4.1 

(( 

+  + 

+ 

+ 

+ 

no 

variation. 

From  inspection  of  this  table,  we  see 

1*.  So  long  as  the  value  substituted  for  x  is  less  than  the 
least  root  of  the  functions  there  is  no  change  in  the  signs. 

2°.  When  in  the  process  of  the  substitution  Vj,  V3  vanish, 
the  adjacent  functions  are  of  opposite  signs,  and  no  change 
takes  place  in  the  number  of  the  variations. 

3®.  That  whatever  changes  of  sign  take  place  in  the  secondary 
functions  Vi,  &c.,  the  number  of  variations  is  not  affected  by 
these  changes. 

4°.  That  in  every  case  when  the  primitive  function  changes 
sign,  there  is  a  loss  of  one  variation  and  of  one  only. 

5°.  The  roots  of  the  equation  being  comprised  within  the 


GENERAL   THEORY   OF   EQUATIONS.  307 

limits  of  the  values  assigned  to  :c,  the  number  of  variations 
lost  is  precisely  equal  to  the  number  of  the  real  roots  of  die 
equation. 

250.  If  the  number  only  of  the  real  roots  is  required,  it  will 
be  sufficient  to  substitute  -|-  oo,  and  —  QO,  the  extreme  limits, 
instead  of  x, 

Ex.  1.  Required  the  number  of  real  roots  in  the  equation 
a:3_92,2_|_23a;_i5==o. 
We  shall  have  for  the  functions 

V==    a?-   9a;2  +  23a;— 15' 
Vi  =  3a:2— 18a:+23 
V2=    a;  -  3 

V3=:4.    . 

Substituting  in  these  functions  -j-  oo  and  —  oo  successively, 
we  have  the  following  results  : 

a:=  4-  00  gives  -f-  -|-  -j-  -}-  No  variations. 

xz=. —  00     " 1 1-     3  variations. 

Here  the  difference  of  the  variations  is  3.  There  will  be, 
therefore,  three  real  roots  in  the  proposed  equation.  And  as 
the  equation  contains  no  permanence,  the  three  roots  will  all  be 
positive. 

If  the  situation  as  well  as  number  of  the  roots  is  required,  we 
proceed  as  follows.     Beginning  with  0,  we  substitute  the  series 
of  natural  numbers  1,  2,  3,  4,  &c.,  for  x  in  the  functions,  thus, 
V  Vi  V^  V3 


a:  =  0  gives 1 \- 

3  variations. 

x  =  l     " 

0+-  + 

2 

a:  =  2     " 

+ + 

2 

x  =  S     " 

0  —  0   + 

1  variation. 

a:  =  4    " 

— +  + 

1 

x  =  b     " 

0+  +  + 

no  variation. 

Since  the  numbers 

1,  3,  5,  reduce 

the  proposed 

to  0,  these 

are  the  roots  of  the  et 

luation,  which  - 

are  thus  completely  deter- 

mined  by  the  process. 

308 


ELEMEJiTS   OF   ALGEBRA. 


Ex.  2.  To  find  the  number  and  initial  figures  of  the  roots  of 
the  equation  a;^  —  4:x'^  —  6x-\-8=0,  « 

The  functions  are  V  =      x^  — 4^x^  —  6 x-\-& 

Vi=   3a:2_8a:— 6 

V2=17a:  —12 

The  sign  only  of  Vg  is  written,  since  this  is  all  that  is  neces- 
sary, and  this  may  be  determined  without  actually  performing 
the  division. 

Substituting  +  c»,  and  —  oo, 

x  =  -\-<X)  gives  -f-  4"  +  ~f"  ^^  variations. 
x  =  —  00      " 1 [-3  variations. 

There  will  be,  therefore,  three  real  roots  ;  and  as  there  is  one 
permanence  in  the  proposed,  one  of  these  will  be  negative. 

To  determine  the  situation  of  the  roots,  we  substitute  the 
series  of  natural  numbers  0,  1,  2,  &c.,  positive  and  negative, 
thus: 


X  =  0  gives  -\ 

'2Vs 

7ar. 
2 

1 

a:  =  2     " 



1 



1 

1 
0 

xz=      0  gives  -j f- 

x  =  ^l     «     +  +  -  + 

X=:—2      » 1 }- 


Van 
2 
2 
3 


From  the  column  of  variations,  it  is  evident  that  the  roots  lie 
between  0  and  1,  4  and  5,-1  and  —  2.  The  initial  figures 
of  the  roots  are,  therefore,  0,  4,  and  —  1.  To  obtain  the  first 
decimal  figure  of  the  root  between  0  and  1,  we  substitute  in 
the  order  of  tenths.  And  since  1  is  very  near  the  root  we  begin 
with  .9.     Thus 

x=l  gives 1-  4"  1  variation. 

xsss.9    "     -\ h+^  variations. 

The  initial  figures  of  the  roots  are,  therefore,  .9,  4  and  —  1. 

Ex.  3.  How  many  real  roots  has  the  equation  a^  —  6x^-j- 
11 X  —  6  =  0  »  Ans.  Three,  viz.,  1,  2,  and  3, 


GENERAL  THEORY  OF  EQUATIONS.  309 

Ex,  4.  How  many  real  roots  has  the  equation  a:^  —  5  a:^  -|" 
6a;— 1=0?  Ans.  1. 

Ex.  5.  How  many  real  roots  has  the  equation  3^  —  7  a:  + 
7  =  0? 
Ans.  3.  Two  between  1  and  2,  and  one  between  —  3  and  —  4. 


SECTION   XXVII.  —  Solution  of  numerical  equations  of 

ANY   DECREE. 

The  principles  now  obtained  are  sufficient  for  the  solution  of 
numerical  equations  of  any  degree  with  one  unknown  quantity. 
We  proceed  to  apply  them. 

commensurable  roots. 

251.  A  commensurable  root,  it  will  be  recollected,  is  one 
which  has  a  common  measure  with  unity,  h  may,  therefore, 
be  an  integer,  or  a  definite  fraction,  such  as,  \,  ^,  ^,  &c.  We 
commence  with  the  research  for  integral  roots. 

Since  every  root  of  an  equation  (art.  229)  is  a  divisor  of  the 
last  or  absolute  term,  the  integral  roots  of  equations  must,  as  we 
have  seen,  be  sought  among  the  entire  divisors  of  this  term. 
They  will  be  comprised,  moreover,  between  the  limits  of  the 
roots.  Every  equation,  the  coefficients  of  which  are  entire 
numbers  and  that  of  the  first  term  equal  to  unity  must  have 
(art.  230)  entire  numbers  for  its  commensurable  roots.  We 
begin  with  equations  of  this  description. 

Ex.  1.  Let  it  then  be  proposed  to  find  the  integral  roots  of 
the  equation  ar^  —  5  a;*+  ar^  -f- 16  a:^  —  20  a:  -)-  16  =  0. 

The  limits  of  the  roots  are  5  and  —  4.  The  integral  roots 
of  the  equation  must,  therefore,  be  found  among  the  entire  divi- 
sors of  16  comprised  between  5  and  —  4.  These  are  4,  2,  1, 
—  1,  — 2,  —  4.     But  1  and  —  1,  it  will  be  seen  at  once,  are 


310 


ELEMENTS    OF   ALGEBRA. 


not  roots,  since  they  will  not  satisfy  the  equation.     The  trial 

divisors,  are,  therefore,  reduced  to  4,  2,  —  2,  —  4.     These  we 

try  in  sucpession. 

1-5  +  1  +  16  —  20+16 
+  4-4—12+16—16 

_1_3+    4—   4 

2  +  2—   2+   4 


—  2 


1_1+   2 

2  +  2—   2 


1-1  +  1, 

From  the  operation,  4,  it  is  evident,  is  a  root,  since  the  divis- 
ion by  a:  —  4  leaves  no  remainder.  Dividing  the  quotient  of 
this  division  by  a:  —  2,  there  is  no  remainder ;  2  is,  therefore,  a 
root.  Dividing  next  this  last  quotient  by  x-\-2,  there  is  no 
remainder ;  hence  —  2  is  a  root.  We  proceed  no  further  with 
the  negative  divisors,  since  the  equation  exhibiting  but  one  per- 
manence can  have  but  one  negative  root,  and  that  is  now  found. 
The  coefficients  of  the  remainder  after  these  successive  divisions 
are  1  —  1  +  1.^  Hence  we  shall  have  fpr  the  depressed  equa- 
tion containing  the  remaining  roots  of  the  proposed,  x^  —  a:  +  1 
==  0,  the  roots  of  which  are  imaginary. 

Ex.  2.  What  are  the  integral  roots  of  the  equation 
a;3__i0a;2  +  31ar  — 30  =  0? 

Since  the  equation  exhibits  only  variations  of  sign  there  can 
be  no  negative  roots.  It  will  be  necessary,  therefore,  to  find 
only  the  superior  limit  of  the  positive  roots,  and  to  employ  the 
divisors  of  30  between  0  and  this  limit.  The  limit  is  10 ,  and 
since  1  is  obviously  not  a  root,  the  trial  divisors  will  be  2,  3,  5, 
and  6,  by  means  of  which  we  obtain  2,  3,  and  5,  for  the  roots. 
And  the  equation  being  of  the  third  degree  only,  these  are  all 
the  roots,  and  the  equation  is  completely  solved. 

Ex.  3.  What  are  the  integral  roots  of  the  equation  x^  —  10 
a,-5  +  37a;2  — 60a;  +  36==0? 

The  equation  has  no  negative  roots.  Applying  the  process 
we  obtain  2  and  3  for  the  positive  roots.     These,  it  is  evident^ 


GENERAL  THEORY  OF  EQUATIONS.  311 

may  be  equal  roots.  To  discover  whether  this  is  the  case,  we 
continue  the  division  by  2  and  3  upon  the  depressed  equation 
after  taking  out  the  roots  2  and  3.  Or  we  substitute  2  and  3 
successively  in  the  derivative  of  the  proposed.  And  since  they 
satisfy  this  equation  also,  they  are  (art.  247)  equal  roots.  Thus 
the  integral  roots  of  the  proposed  are  2,  2,  3,  3  ;  and  since  it  is 
of  the  fourth  degree  only,  these  are  all  its  roots. 

252.  When  the  number  of  trial  divisors  is  large,  the  process 
is  laborious.  It  will  admit  of  some  simplification  in  the  manner 
we  will  now  explain. 

Let  it  be  proposed,  for  example,  to  find  the  integral  roots  of 
the  equation  a^  —  I6x^ +  14:X  —  120  =  0. 

The  equation  has  no  negative  roots.  The  superior  limit  to 
the  positive  roots  is  15 ;  and  since  1  is  obviously  not  a  root,  the 
trial  divisors  will  be  12,  10,  8,  6,  5,  4,  3,  2. 

In  making  trial  of  the  divisors  it  will  be  found  most  conven- 
ient to  invert  the  order  of  the  coefficients,  ffiat  is,  arrange  with 
reference  to  the  ascending  powers  of  x,  and  then  change  the 
signs  of  the  divisor,  the  effect  of  which  will  be  merely  to  change 
the  signs  of  the  quotient. 

Thus,  in  the  present  example,  if  we  begin  with  3,  the  work 
will  be 


_  120 +  74— 15+1 
—  40 


34. 

3  is  not  a  root,  since  34  is  not  divisible  by  3,  and  the  total  divis- 
ion by  3  is,  therefore,  impossible.     Try  next  4. 

-120  +  74-15  +  1 
_30  +  ll_l 

44—4      0. 

4  is  a  root,  since  there  is  no  remainder.     Proceeding  with  the 
process  we  find  5  and  6  for  the  remaining  roots. 

If  the  operations  performed  above  be  examined  with  attention, 
the  following  principle,  which  may  easily  be  proved  to  be 
general,  will  be  discovered,  viz. :  In  order  that  a  number  may 
be  a  root,  the  first  coefficient  or  absolute  term,  must  be  divisible 


312  ELEMENTS   OF   ALGEBRA. 

by  it,  so  must  also  the  sum  of  the  quotient  and  the  next  co- 
dfficient,  the  sum  of  this  last  quotient  and  the  next  coefficient, 
and  so  on  throughout,  the  last  quotient  being  always  —  1. 
Any  number  which  will  not  sustain  these  tests  in  succession  is 
not  a  root. 

Taking  advantage  of  what  has  now  been  said,  the  entire 
work  in  the  example  proposed  may  be  conveniently  performed 
as  follows : 


-120 

12, 

10, 

8,         6,        5,        4, 

3, 

2, 

+   74 

-10,- 
64, 

-12,- 
62, 

.  15,  _  20,  — 24,  — 30,- 
59,       54,      50,      44, 

-40,- 
34, 

-60, 
14, 

—    15 

^ 

# 

^         9,       10,       11, 
_6,_   5,-    4, 

# 

7 
—  8 

+      1 

-1,-    1,-    1, 

• 

—  4 

—  3 

The  coefficient*  of  the  proposed  are  placed  in  a  vertical 
column  with  their  proper  signs.  On  the  same  horizontal  line 
with  the  first  coefficient,  or  absolute  term,  are  arranged  the  trial 
divisors.  Beneath  these,  in  the  same  horizontal  line  with  the 
second  coefficient,  are  placed  the  quotients  arising  from  the 
division  of  the  absolute  term  by  the  trial  divisors.  Beneath 
these,  in  the  next  line,  are  placed  the  sums  of  the  first  quotient 
and  second  coefficients.  These  are  the  dividends  to  be  divided 
next  by  the  trial  divisors  ;  and  the  quotients  of  this  division  are 
placed  beneath  them  in  the  next  line  below,  and  in  the  same 
line  with  the  third  coefficient,  and  so  on  in  order. 

On  the  second  division  the  divisors  12,  10,  8,  and  3  are  re- 
jected, and  the  divisor  2  on  the  fourth.  The  only  divisors  which 
sustain  the  required  tests  are  6,  5,  and  4,  and  these  are,  there- 
fore, roots  of  the  equation. 

The  following  examples  will  serve  as  an  additional  exercise. 

Ex.  1.  What  are  the  integral  roots  of  the  equation  a^  -\-3 
a:2_8a;4_iO  =  0?  Ans.  — 5. 

Ex.  2.  What  are  the  integral  roots  of  the  equation  x^  -\-  4: 
a^^x'—\Qx  —  l2  =  0'i         Ans.  2,  — 1, —  2,  and —3. 


GENERAL   THEORY    OF   EQUATIONS.  313 

Ex.  3.  What  are  the  integral  roots  of  the  equation  x^  —  a:* 
—  13a:2_|_i6a._4Q^0?  Ans.  4,and— 4. 

Ex.  4.  What  are  the  integral  roots  of  the  equation  x^  —  7 
d?J^nx^-^Y\x-\-^^^%  Ans.  1,  l,2,and3. 

253.  The  method  above  is  applicable  also  to  equations,  the 
coefficients  of  which  are  entire  numbers  and  that  of  the  first 
term  difierent  from  unity.  In  this  case  the  last  quotient  will 
be  —  1  multiplied  by  the  coefficient  of  the  first  term. 

Ex.  1.  Find  the  integral  roots  of  the  equation  2  a:^ — 22  a;'' 
4-62a:  — 42  =  0.  Ans.  1,  3,  and  7. 

Ex.  2.  What  are  the  integral  roots  of  the  equation  3  a;^  —  23 
a:2_|_44a,_2o==0?  Ans.  2  and  5. 

254.  We  proceed  next  to  equations  which  contain  fractional 
commensurable  roots. 

Let  it  be  proposed,  for  example,  to  find  the  commensurable 
roots  of  the  equation 

To  find  the  integral  roots,  we  free  the  equation  from  denomi- 
nators, which  will  not  alter  the  value  of  the  roots.  This  gives 
24a;*_  74:c3  +  61  a;2  —  19  a:  +  2  =  0. 

Applying  next  the  process  above  to  this  equation  we  obtain 
2  for  the  integral  root.  "  Removing  this  from  the  proposed,  the* 
reduced  equation  will  be 

Since  2  is  the  only  integral  root  of  the  proposed,  the  commen- 
surable roots  of  this  last,  if  it  have  any,  must  be  fractional. 
To  determine  these  we  transform  the  equation  into  another 
whose  roots  shall  be  integral.     In  order  to  this  (art.  240,  no.  2) 

X  . 

we  put  a;  =  — ,  24  being  the  least  common  multiple  of  the  de- 
nominators, and  we  have  for  the  transformed 

a:s  _  26  a:2  +  2 16  a:  —  576  =  0.     (3) 
Applying  the  process  for  integral  roots  to  this  equation  the 
27 


314  ELEMENTS   OF   ALGEBRA. 

roots  are  found  to  be  12,  8,  and  6.  But  these,  it  is  evident, 
are  24  times  larger  than  those  of  equation  (2).  Hence  the 
roots  of  (2)  are  ^,  ^,  and  ^. 

The  roots  of  the  proposed  equation  will  be,  therefore,  2,  ^,  ^, 
and  ^. 

Ex.  2.  Find  the  commensurable  roots  of  the  equation  a?  — 
31  1  1 

3Q^'  +  3^^  -  30=^^-  ^"s-  h  h  and  f 

Ex.  3.  Find  the  commensurable  roots  of  the  equation  3  a:^  — 
14  a:2  +  21  a;  -  10  =  0.  Ans.  1,  |,  and  2. 

Ex.  4.  What  are  the  commensurable  roots  of  the  equation 
12a;*4-20a;3- lla:2-5a;  +  2  =  0? 

Ans.  ^,  ^,  —  I,  and  —  2. 

INCOMMENSURABLE    ROOTS. 

255.  Incommensurable  Eoots  are  those  which  have  no  com- 
mon measure  with  unity,  such  as  surds  and  interminable  deci- 
mals. 

We  proceed  next  to  equations  having  incommensurable  roots. 
These  may  be  found  by  the  principles  explained  above  to  any 
degree  of  approximation  we  please. 

Let  it  be  proposed,  for  example,  to  find  the  roots  of  the  equa- 
tion a:^  _|.  ^  ^2  _  io2  z  +  181  =  0. 

The  substitution  of  +  oo  and  —  00  shows  that  the  equation 
has  three  real  roots.  And  since  there  is  but  one  permanence, 
two  of  the  roots  will  be  positive.     The  functions  are 

V=        a:3_[_ii^_io2a:+181 
Vi==      3a:2-t-22a:  — 102 
¥2=  122  a;  — 393 

Putting  a;  =  0  gives  -| \-    2  variations. 

a:=l     "     + + 

a;  =  2     "     -| (- 

a;  =  3     «      -j [-     2  «         - 

2:  =  4     "     -f-  -|-  _|-  ._j_     no  variation. 


GENERAL  THEORY  OF  EQUATIONS.  315 

The  two  positive  roots  are,  therefore,  comprised  between  3 
and  4.  To  approach  them  more  nearly,  we  transform  the  pro- 
posed equation  into  another  whose  roots  shall  be  less  by  3. 
The  functions  will  be, 

V=        a^^20x^  -Qx+l 

Vi=     Sz'-\-4S)x-9 

V2=122a;  — 27 

V3  =  +.  ■  . 

Putting  in  these 

a;  =  0  gives  -\ \-    2  variations. 

x  =  .l     «     + + 

x  =  .2    «     -| 1-    2  « 

x  =  .3     "     +  -h  "t"  -f*    ^°  variation. 
The  two  positive  roots  of  the  reduced  equation  are  comprised, 
therefore,  between  .2  and  .3.     Hence  those  of  the  proposed  are 
comprised  between  3.2  and  3.3. 

To  approach  the   roots  still   nearer  we  transform  the  last 
transformed  equation,  a^  -\-20  x^  —  9  a;  -f-  1>  into  another,  the 
Toots  of  which  shall  be  less  by  .2.     The  functions  will  then  be 
V  =        u^ -\-20.6 x^  —  .S8x -}-  .008 
\\=     3x'  +  4.l.2x^,Q8 


Va=122a:  — 2.6 

v,=+. 

In  these  ,x=  0    gives  -| \- 

2  variations. 

x  =  m     "     + f- 

2 

a;  =  .02     « 1- 

1  variation. 

^  =  .03     "     +  +  +  + 

no  variation. 

The  two  positive  roots  of  the  last  transformed  are  comprised, 
therefore,  hetween  .01,  .02  and  .02,  .03.  By  consequence  the 
first  three  figures  in  those  of  the  proposed  are  3.21,  and  3.22 ; 
and  since  the  sum  of  the  three  roots  is  —  11,  the  negative  root 
will  be  —  17.43.  We  shall  have,  therefore,  for  the  approximate 
roots  of  the  proposed  3.21,  3.22,  and  —  17.43. 

By  the  process  above  we  may  approach  the  roots  of  an  equa- 
tion as  nearly  as  we  please.     The  process  is,  however,  laborious, 


316  ELEMENTS    OF    ALGEBRA.  * 

and  may  be  much  abridged.  Th^  following  method  of  accom- 
plishing this  object  was  first  published  by  W.  G.  Horner,  of 
Bath,  England,  in  1819. 

Horner's  method  of  approximation. 

256.  This  method  is  founded  upon  the  following  principles. 

Let  there  be  the  equation 

V==a;'^  +  Aa;"-i+ Ta:  +  U  =  0. 

Let  x'  be  the  part  of  the  root  of  this  equation  already  found, 
and  y  the  remaining  part,  y  being  very  small  compared  with  x'. 
Then,  transforming  the  proposed  into  another,  the  roots  of  wjiich 
shall  be  less  by  x\  we  obtain 

V'  =  2/"  +  A'2/"-^+   .   .*.    .   r2/  +  U'  =  0; 
then,  since  2/  is  a  very  small  quantity,  all  the  terms  of  this  equa- 
tion, in  which  the  power  of  y  is  above  the  first,  may  be  neglected, 
and  the  equation  T'  ?/  -(-  U'  =  0,  will  give  the  value  of  y  very 
nearly.     Resolving  this  equation  we  have 

U' 

2/  =  — ^• 

Thus,  in  the  equation  a:^  —  5  o;^  -)-  8  a:  —  1  =  0,  the  value  of 
a:,  it  is  easy  to  see,  lies  between  0  and  1.  Neglecting  the  terms 
which  involve  x  above  the  first  power,  we  have  for  a  trial  equa- 
tion 8  a;  —  1  =  0,  from  which  we  obtain  x  =  .125.  The  first 
figure  of  this  decimal  is  true,  since  by  substitution  it  will  be 
found  that  the  value  of  x  lies  between  .1  and  .2.  Thus  the 
first  figure  of  the  approximate  root  is  easily  determined.  In 
order  to  proceed  to  the  next,  we  transform  the  equation  V  =  0 
into  another  whose  roots  shall  be  less  by  the  part  last  found. 
This  will  give  a  new  trial  equation,  by  which  to  find  the  second 
figure  of  the  root,  and  so  on.     The  rule  may  be  thus  stated  : 

1°.  Find  by  trial,  or  by  Sturm's  Theorem,  the  situation  and 
first  figure  of  the  real  roots. 

2°.  Transform  the  proposed  equation  into  another,  whose 
roots  shall  be  less  than  those  of  the  given  equation  by  the  part 
of  the  root  already  found. 


GENERAL  THEORY  OF  EQUATIONS.  317 

3°.  With  the  absolute  term  in  this  transformed  equation  for 
a  dividend,  and  the  coefficient  of  x  for  a  divisor,  find  the  next 
figure  of  the  root  and  verify  it  in  the  transformed  equation. 

4°.  Diminish  the  roots  of  the  transformed  equation  by  the 
value  of  the  figure  last  obtained,  divide  as  before  for  the  next 
figure,  and  so  on. 

Ex.  1.  Find  the  roots  of  the  equation  y?  -\- \(i  y?  -\- h  x  — 
260  =  0.     The  functions  will  be 

Vi  =  3a;2-j-20a;  +5 
¥3=  17a; +  239 

V3  =  -. 

Substituting  in  these  functions  as  above,  the  equation,  it  will 
be  found,  has  but  one  real  root,  the  first  figure  of  which  is  4. 
Transforming  it  into  another,  the  roots  of  which  shall  be  less 
by  4,  the  operation  will  be 

1  I  1         +10        4-5        —260 
4|  4  56  244 

14  61  -16 

_4  J72 

18  133 

22 

and  the  transformed  will  be  a:^  +  22  a:^  -f-  133  a;  —  16  =  0.  (2) 
We  have,  therefore,  for  the  first  trial  equation  133  a;  —  16 
=  0,  which  gives  .1  for  the  second  figure  of  the  root.  Thus 
the  root  of  the  proposed  will  be  4.1  nearly.  Transforming 
next  the  equation  (2)  into  another,  whose  roots  shall  be  less  by 
.1,  the  operation  will  be 

1|1        _i.22.  +133  —16 

.1  I  0.1  2.21  13.521 

22.1  135.21         -2.479 

.1  2.22 


22.2  137.43 

.1 


22.3 
27* 


318  ELEMENTS   OF   ALGEBRA. 

and  the  transformed  will  be  a:^  +  22.3  a;2_|- 137.43^  —  2.479 
=  0.     (3) 

We  have,  therefore,  for  the  next  trial  equation,  137.43  a:  — 
2.479  =  0,  which  gives  .01  for  the  next  figure  of  the  root, 
The  root  will  be,  therefore,  4.11  nearly.  Transforming,  next, 
equation  (3)  into  another,  whose  roots  shall  be  less  by  .01,  the 
operation  will  be 


1 

.01 

1 

+  22.3 

.01 

+  137.43 
.2231 

—  2.479 
1.376531 

22.31 
.01 

137.6531 
.2232 

— 1.102469 

22.32 
.01 

137.8763 

22.33 

and  the  transformed  will  be 

2^  +  22.33  x^  +  137.8763  x  — 1.102469  =  0.     (4) 
And  we  have   for  the  next  trial  equation   137.8763  x  — 
1.102469  =  0,  by  which  we  obtain  .007  for  the  next  figure  of 
the  root.     Transforming  (4)  into  another,  whose  roots  shall  be 
less  by  .007,  the  operation  will  be 


1 

.007 

1 

+  22.33 
.007 

H 

•  137.8763 
.156359 

- 1.102469 
.966228613 

22.337 
.007 

138.032659 
.156408 

-  .13624038^ 

22.344 
.007 

138.189067 

22.351 

and  the  transformed  will  be 

a^  +  22.351  a:^  +  138.189067  x  -  .136240387  =  0,  (5) 
and  the  next  trial  equation  is  138.189067  a;  —  .136240387  =  0, 
from  which  we  obtain  .0009  for  the  next  figure  of  the  root. 
The  root  of  the  proposed  will  be,  therefore,  4.1179  nearly.  In 
like  manner  the  approximation  may  be  pushed  as  far  as  we 
please. 


GENERAL  THEORY  OF  EQUATIONS. 


31» 


The  calculations  may  be  performed  more  concisely  as  in  the 
following  table : 


1 

4 

1  +10 
4 

56 

-  260     1  4.11 
244 

14 
4 

61 

72 

^-16 
13.521 

18 
4 

«=133 
2.21 

#_  2.479 
1.376531 

1. 
.1 

=^22.1 
.1 

135.21 
2.22 

=^  —  1.102469 

.966228613 

22.2 
.1 

^  137.43 
.2231 

«=  — .136240387 

1 
.01 

=^22.31 
.01 

137.6531 
.2232 

22.32 
.01 

^  137.8763 
.156359 

1 
.007 

^  22.337 

7 

138.032659 
.156408 

22.344 

7 

=^  138.189067 

=^22.351 

The  process,  when  compared  with  the  previous  work,  will  be 
easily  understood.  The  coefficients  of  the  successive  trans- 
formed equations  are  marked  with  a  star.  The  .1  placed  at 
the  right  of  =^22  in  the  first  column  is  the  .1  added  in  the  first 
step  of  the  process  for  obtaining  the  second  transformed  equa- 
tion, the  addition  being  more  conveniently  made  in  this  manner. 
A  similar  remark  applies  to  the  right  hand  figure  of  the  other 
coefficients  in  the  same  column. 

The  successive  figures  may  be  verified  as  they  are  obtained 
in  the  transformed  equation.  The  process,  it  is  evident,  be- 
comes more  accurate  as  we  proceed.  After  four  or  more 
decimals  have  been  obtained,  two  or  three  more  may  in  general 
be  found  by  simple  division. 

The  sign  of  the  last  term  will  sometimes  change  in  the 
course  of  the  operation.    Unless  there  is  in  this  case  a  change 


320 


ELEMENTS    OF   ALGEBRA. 


of  sign  in  the  preceding  column  also,  the  figure  which  has 
given  rise  to  the  change  must  he  incorrect.  This  change  may 
not,  however,  always  occur  at  the  same  figure  of  the  root. 

257.  In  the  preceding  example  a  greater  number  of  decimal 
places  has  been  employed  than  is  necessary  to  obtain  the  root 
to  the  degree  of  approximation  attained,  and  the  work  might 
have  been  abridged  by  the  omission  of  some  of  them. 

Let  it  be  required,  as  a  second  example,  to  find  the  roots  of 
the  equation  7^  —  \1  x^  -\-b^z  —  350  =  0,  to  three  places  of 
decimals. 

The  equation  has  but  one  real  root,  the  first  two  figure's  of 
which,  found  by  trial,  or  Sturm's  Theorem,  are  14.  The  re- 
mainder of  the  work,  according  to  the  rule,  will  be  as  follows : 


1 

14 

1   —17 
14 

+  54 
—  42 

—  350  1  14.954 
168 

-  3 
14 

12     =^- 
154 

-182 
170.379 

11 
14 

=^25.9 
.9 

26.8 
.9 

=^166     ^. 
23.31 

189.31 
24.12 

=^  213.413 

1.3|875 

-  11.621 
10.740 

875 

1 
.9 

=^  —  880 
865 

125 
275664 

— 14  1  849336 

1 

.05 

^217.75 
5 

214.8  175 
1.3  900 

27.80 
5 

^216. 

2075 
111416 

1. 

.00^ 

^27.854 
\                      4 

27.858 
4 

216. 

318916 
111432 

216. 

430348 

27.862 

If  this  work  is  examined  with  attention,  it  will  be  seen  that 
the  result  will  still  remain  the  same,  if  all  the  figures  to  the 
right  of  the  vertical  lines,  and  those  below  the  vertical  line  in 


GENERAL   THEORY   OF   EQUATIONS. 


821 


the  left  hand  column  are  omitted,  by  which  the  labor  will  be 
much  abridged. 

The  object  is  to  retain  no  more  decimal  places  in  the  last 
column  than  are  necessary  ;  and  these,  in  general,  will  not  be 
greater  than  the  number  of  places  required  in  the  root.  Thus, 
in  the  present  example,  three  places  only  being  required  in  the 
root,  we  cut  off  in  the  last  column,  by  a  vertical  line,  all  the 
remaining  figures  after  three  places  have  been  obtained.  This 
occurs  after  the  operations  with  the  figure  9  of  the  root  have 
been  completed.  Then,  since  the  multiplication  by  each  new 
figure  of  the  root  produces  one  new  decimal  place  in  each 
column  as  we  proceed  from  left  to  right,  in  order  that  no  new 
decimal  places  may  occur  in  the  right  hand  column  we  must, 
it  is  evident,  cut  off  one  figure  in  the  column  next  preceding, 
two  figures  in  the  column  next  preceding  that,  and  so  on.  The 
operation  with  the  figure  5  of  the  root  being  completed,  we 
again  cut  off  one  figure  in  the  column  next  preceding  the  last, 
and  two  in  the  column  next  preceding  that,  and  so  on.  The 
work  will  then  stand  thus : 


1 

14 

-17 
14 

54 

-42 

—  350     14. 
168 

-3 
14 

12 
154 

^—182 
170.379 

11 
14 

=^^166 
23.31 

*— 11.621 
10.741 

1 
.9 

^«=25.9 
9 

189.31 
24.12 

—  .880 
865 

26.8 
9  . 

*  213.4 
1.3 

3 

8 

-15 
15 

1 

.05 
1 
.004 

#2 

7.7 

1 

214.8 
1.3 

1 
9 

1-1.21 

7.8 

2|1|6|.2 

Care,  it  is  evident,  must  be  taken  that,  in  the  result  of  each 
operation,  the  figure  immediately  preceding  the  one  cut  off 
remain  the  same  as  if  the  contraction  were  not  made.  Thus 
=  0?  21 


322  ELEMENTS   OF   ALGEBRA. 

in  the  operation  with  the  figure  5  of  the  root,  we  continue  the 
use  of  the  77,  cut  off  in  the  left  hand  column,  until  the  operation 
with  the  5  is  completed.  We  retain  also,  in  the  column  next 
following,  one  of  the  decimal  places  cut  off.  This  gives,  when 
the  operation  with  the  5  is  completed,  216.2  in  this  column, 
precisely  as  if  the  contraction  were  not  made.  Cutting  off  next 
one  figure,  the  2,  in  the  column  before  the  last,  and  two  in  the 
column  before  that,  there  will  be  none  remaining  in  this  column, 
and  the  multiplication  by  4,  the  next  figure  of  the  root,  will 
produce  no  effect  upon  the  following  columns. 

Having  obtained  the  4,  two  additional  figures  may  be 'found, 
by  simple  division.  In  order  to  this,  cutting  off  the  6  in  the 
column  before  the  last,  we  have  21  for  a  divisor  and  15  for  a 
dividend,  which  gives  0  for  the  next  figure  of  the  root.  Again 
cutting  off  the  1  in  the  column  before  the  last,  we  have  2  for  a 
divisor  and  15  for  a  dividend,  which  gives  7  for  the  next  figure 
of  the  root;  and  the  operation,  true  to  5  places  of  decimals,  is 
now  terminated. 

There  is  room  for  the  exercise  of  judgment  in  the  use  of  the 
figures  cut  off.  The  object  being  to  keep  the  right  hand  figure 
in  the  result  of  each  operation  what  it  would  be  if  the  contrac- 
tion were  not  made,  the  learner  must  judge,  in  each  case,  what 
is  necessary  for  this  purpose.  What  has  been  done  will  serve 
as  a  general  direction.  After  a  little  practice  the  operations 
will  be  easily  executed,  and  the  root  obtained,  to  any  degree  of 
approximation  required,  with  extreme  facility. 

Ex.  3.  To  find  the  roots  of  the  equation  oi?  —  7  a;  -[-  7  =  0. 
There  will  be  one  negative  and  two  positive  roots.  To  find  the 
negative  root,  we  change  the  sign  of  the  alternate  terms,  and 
proceed  as  for  a  positive  root.  The  result,  with  its  sign  changed, 
will  be  the  negative  root  sought. 

The  yots  are  1.356895,  1.692021,  and -- 3.048917,  true  to 
six  places  of  decimals. 

Two  of  the  roots  being  obtained,  the  other  may  be  found  by 


GENERAL  THEORY  OF  EQUATIONS.  323 

the  principle,  art.  229.     Thus,  to  find  the  negative  root,  we  take 
the  sum  of  the  two  positive  roots  with  the  contrary  sign. 

Ex.  4.  Find  one  root  of  the  equation  a^-^-Sx^-^bx—  178 
=  0,  true  to  6  places  of  decimals.  Ans.  4.538825. 

Ex.  5.  Find  the  roots  of  the  equation  3^  —  5x  —  3  =  0, 
trae  to  four  places  of  decimals. 

Ans.  2.4908,  -  0.6566,  -  1.8342. 

Ex.  6.  Find  a  root  of  the  equation  3^-\-2  x^  —  23  a;  —  70  = 
0,  true  to  four  places  of  decimals.  "  Ans.  5.1345u 

Ex.  7.  Find  the  root  of  the  equation  x^-\-Sx* '}-2x^ —  3  x^ 
—  2x  —  2  =  0,  to  four  places  of  decimals.         Ans.  1.0591. 

Ex.  8.  Find  the  roots  of  the  equation  x*  —  x^  -\-2x^  -\-  x  —  4 
=  0.  Ans.  1.14699459,  and  —  1.0905935. 

258.  The  preceding  process  may  he  applied  to  the  extraction 
of  the  roots  of  numbers. 

Ex.  1.  Let  it  be  required  to  extract  the  third  root  of  9. 

In  this  case,  we  have  to  solve  the  equation  a:^  —  9  =  0, 
which,  by  the  preceding  process,  gives  2.0800838  for  the 
answer. 

Ex.  2.  To  find  the  roots  of  the  equation  x"^  —  2=0. 

Ans.  ±  1.414213. 

Ex.  3.  Find  the  fifth  root  of  2.  Ans.  1.148699. 

SECTION  XXVIII.     Elimination.  —  Solution  of  equations 

WITH   two    or   more    UNKNOWN   QUANTITIES. 

259.  The  equations  of  any  degree,  thus  far  solved,  contain 
one  unknown  quantity  only.  We  proceed  next  to  equations 
with  more  than  one  unknown  quantity. 

Let  there  be  two  equations  with  two  unknown  quantities.  It 
is  proposed  to  find  the  systems  of  values  for  the  unknown 
quantities  x  and  y,  that  will  satisfy  these  equations.  In 
order  to  this,  we  must  first  eliminate  one  of  the  unknown  quan- 
tities. 

Let  the  equations,  arranged  in  reference  to  a:,  be  represented 
by  A  =  0,B  =  0. 


324  ELEMENTS   OF   ALGEBRA. 

Applying  to  these  the  principle  of  the  greatest  common  divisor, 

let  Q  =  the  quotient  of  A  by  B,  and  R  =  the  remainder ;  then 

A  =  BQ  +  R. 

It  follows  from  this  equality,  that  all  the  values  of  x  and  y, 
which  give  A  =  0,  B  =  0,  must  also  give  R  =  0.  The  sys- 
tem of  equations  A  =  0,  B  =  0,  may,  therefore,  be  replaced 
by  the  more  simple  system,  B  =  0,  R  =  0. 

Dividing  next  B  by  R,  let  a  new  remainder  R^  be  reached. 
We  may,  in  like  manner,  substitute  for  B  =  0,  R  =  0,  the  sys- 
tem R  =  0,  R'  =  0,  in  which  R'  is  of  a  lower  degree  in  respect 
to  X  than  R.  And  we  may  thus  continue  until  a  remainder,  R", 
for  example,  is  obtained  independent  of  x.  The  original  sys- 
tem, A  =  0,  B  =  0,  may  then  be  replaced  by  the  system  R'  = 
0,  R"  =  0,  in  which  R"  contains  y  only,  and  R'  is  generally  of 
the  first  degree  in  x.  The  equation  R"  =  0,  from  which  x  is 
eliminated,  is  called  the  JiTicd  equation. 

The  values  of  y  being  obtained  from  the  final  equation,  those 
of  X  will  be  easily  found ;  and  thus  the  systems  of  values  for  x 
and  y^,proper  to  satisfy  the  proposed  equations,  be  determined. 

Ex.  1.  To  find  the  values  of  2:  and  y  in  the  equations 

X  -^y  —   8  =  0. 
Applying  the  process  of  the  greatest  common  divisor, 


x'+{y^8)x 


x  +  y-8 


x  —  y  +  8 

^{y  —  8)x—f  —  M 

—  (y  — 8)0:  — y^-|-16y~64 

2  2/2  _  16  2^  _|_  30  =  Remainder. 
Putting  this  remainder  equal  to  0,  we  have  for  the  final  equa- 
tion, 2^^  —  8^+15  =  0. 

Resolving  this  equation  we  obtain  2/  =  3or5;  and,  substi- 
tuting in  the  divisor,  we  obtain  for  the  corresponding  values 
of  ar,  a:  =  5  or  3.  Thus  the  system  of  values  for  x  and  y, 
which  satisfy  the  proposed  equations,  are  y  =  3,  2:  =  5,  and 
y  =  5,a:  =  3. 


CENERAL  THEOEY  OF  EQUATIONS.  325 

Ex.  2.  Find  the  values  of  x  and  y  in  the  equations, 
^2/* +  2/"  — 333  =  0, 
xf^y—   21  =  0. 
Ans.  y  =  3,  2;  =  2,  or  y  =  18,  a:  =  j^,. 
Ex.  3.  Find  the  values  of  x  and  y  in  the  equations 
4a;  — 22^+    2/^—11  =  0, 
a;  +  42^  —14  =  0. 
Ans.  y  =  3,  a;  =  2,*or  y  =  15,  a:  =  —  46. 

260.  In  the  process  for  finding  the  greatest  common  divisor, 
it  may  be  necessary,  in  order  that  the  division  may  be  exactly 
performed,  to  multiply  one  of  the  quantities  by  a  factor  contain- 
ing y.  In  this  way  roots  may  be  introduced  into  the  final 
equation  foreign  to  the  proposed  equations,  and  which  must  be 
rejected. 

In  like  manner,  for  convenience,  factors  containing  y  may  be 
suppressed  in  the  course  of  the  operations,  which,  when  put 
equal  to  0,  may  give  values  for  y  proper  to  satisfy  the  equations, 
but  which,  from  the  suppression  of  the  factors,  will  not  be  found 
in  the  final  equation.  We  must,  therefore,  in  order  to  a  com- 
plete solution,  make  the  factors  introduced  or  suppressed  equal 
0.  We  shall  thus  establish  relations  which,  with  the  final 
equation,  will  enable  us  readily  to  detect  the  foreign  solutions 
and  to  determine  all  the  values  of  the  unknown  quantities 
proper  to  satisfy  the  proposed  equations. 

261.  Various  simplifications  may  be  introduced  into  the  opera- 
tions, and  the  process  improved  so  as  to  avoid  the  foreign  solu- 
tions. The  general  idea  of  the  process  we  have  given  is  all 
our  limits  admit.     We  subjoin  a  few  additional  examples. 

Ex.  1.  To  find  the  values  of  x  and  y  in  the  equations, 

x^^^xy^r    2/^-1  =  0. 
Ans.  2/  =  —  1,  a:  =  2,  and  y  =:  —  2,  a:  ==  1. 
Ex.  2.  To  find  the  values  of  x  and  y  in  the  equations, 
:^^'iyx^J^{^f-y+\)x-f-\-f-2y  =  ^, 
a^—.2yx-\-f  —  y  =  0,  Ans,     a;  =  2,  7=L 

28 


326  ELEMENTS    OF    ALGEBRA. 

Ex.  3.  To  find  the  values  of  x  and  y  in  the  equations, 

Ans.  The  equations  are  incompatible. 
It  will  be  easy  to  see  how  we  are  to  proceed  with  three  equa- 
tions with  three  unknown  quantities,  and  so  on. 

SECTION.  XXIX.    Infinite  Series. 

262.  An  infinite  series  is  one  in  which  the  number  of  terms 
is  UTilimited ;  the  law  of  the  series  being  generally  discoverable 
by  an  examination  of  a  few  of  the  terms. 

A  converging  series  is  one  whose  successive  terms  decrease, 
or  become  less  and  less. 

A  diverging  series  is  one  whose  successive  terms  increase,  or 
become  greater  and  greater. 

An  ascending  series  is  one  in  which  the  exponents  of  the 
unknown  quantity  continually  increase  ;  and  a  descending  series 
is  one  in  which  the  exponents  continually  decrease. 

When  the  value  of  an  algebraic  expression  cannot  be  exactly 
determined,  we  expand  the  expression  into  a  series,  and  thus 
endeavor  to  obtain  an  approximate  value.  We  have,  therefore, 
two  questions  to  solve  in  respect  to  series.  1°.  To  expand 
algebraic  expressions  into  series.  2°.  To  find  any  term  of  a 
series,  and  the  sum  of  all  the  terms. 

undetermined  coefficients. 

263.  In  the  development  of  algebraic  expressions  in  series, 
the  method  of  undetermined  coefficients  is  found  of  great 
utility. 

We  will  give  a  brief  exposition  of  this  method. 

Let  there  be  the  equation, 

0  =  A  +  Ba;-t.Ca:2-f-Da:*+... 
in  which  the  coefficients  A,  B,  C  .  .  are  independent  of  x;  it  is 
required  to  determine  these  coefficients  so  that  the  equation 


UNDETERMINED  COEFFICIENTS.  *  327 

may  be  true  whatever  value  is  assigned  to  x.  Since  the  coeffi- 
cients A,  B,  C  .  .  are  to  be  determined,  they  are  on  this  account 
called  undetermined  coefficie7its. 

Since  by  hypothesis  the  proposed  equation  must  be  true, 
whatever  value  is  assigned  to  x,  it  must  be  true  for  the  particu- 
lar value  a:  =  0.  Putting  a:  =  0,  the  equation  is  reduced  to 
0  =  A ;  we  have,  therefore,  A  =  0.  Substituting  this  value 
of  A  in  the  equation,  and  dividing  both  sides  by  x,  it  becomes 

0  =  B  +  Ca:  +  Da:2+ .. 
but  since  this  equation  must  also  be  true  whatever  the  value  of 
ic,  putting  a;  =  0,  we   obtain  B  =:  0.     By  the  same  course  of 
reasoning  it  may  be  shown  also  that  C  =  0,  D  =  0. 

If,  then,  we  have  an  equation  of  the  form 

0  =  A  +  Ba;4-Ca:2  +  Da:^+&c., 
in  which  the  coefficients  A,  B,  C,  &c.,  are  independent  of  a:,  in 
order  that  the  equation  may  be  true  whatever  value  is  assigned 
to  X,  each  separate  coefficient  must  necessarily  be  equal  to  zero. 

This  is  the  principle  upon  which  the  method  of  undetermined 
coefficients  is  founded.  We  pass  to  some  applications  of  the 
method. 

Ex.  1.  Let  there  be  a  dividend  x^  —  px-{-p',  let  the  divisor 
be  a;  —  a,  and  the  quotient  x  —  q;  to  determine  the  conditions 
necessary  in  order  that  the  division  may  be  exact. 

Since,  when  there  is  no  remainder,  the  divisor  multiplied  by 
the  quotient  should  produce  anew  the  dividend,  we  must  have 

{x  —  a){x—q)=zx^—px-{-  p\ 
or,  performing  the  multiplication  indicated,  transposing  and  re- 
ducing, 

^  =  {a-\-q—p)x-\'{j^  —  aq)\ 
but  since  this  equation  is  true  whatever  the  value  of  a:,  we  must 
have 

a  -|-  ^  -r— p  =  0,  and  p'  —  a  g  =  0  ; 
eliminating  q  from  these  last,  we  obtain 
(i^=.ap—pf. 

In  order,  therefore,  that  the  division  may  be  exact,  we  must 


328  ELEMENTS   OF   ALGEBRA. 

have  the  relation  c^z=ap  —  p\  or,  which  is  the  same  thing, 
a2  —  ap-{-p'  =  (i. 

Ex.  2.  Let  it  be   proposed  to  decompose  -:; — —-  into 

X" O  X  -j-  o 

fractions,  whose  sum  is  the  given  fraction,  and  whose  denomi- 
nators are  the  factors  of  the  given  denominator. 

The  factors  of  the  denominator  are  x  —  3,  a:  —  2;  we  assume, 

therefore, 

3  +  5a;       _     A  B 


(a;— 3)  (a;  — 2)      a;  —  3   '  x  —  2 

Freeing  from  denominators,  transposing  and  reducing 

0  =  (A  +  B— 5)a:  — (2A  +  3B  +  3; 

whence  A-i-B  =  5,  and2A4-3B  =  --3; 

from  which  we  obtain  A  =  18,  B  =  —  13,  and  we  have 

3  +  5a:      _    18  13 

a:^  — 5a;-|-6       a;  —  3  "~  a;  —  2* 

51x.  3.  Find  for  A  and  B  values,  such  that  we  may  have 

7  +  9a: A  B 

(a;  — 5)  (a;  — 3) ""a;  — 5"^ a;  — 3' 

Ans.  A  =  26,  B=-17. 

Ex.  4.  Find  for  A  and  B  values  such  that  we  may  have 

3a; --5  A  B 

(a:  — 4)(a:— 2)~^a;  — 4      a:  — 2' 

Ans.  A  =  |-,  B  =  J-. 

264.  We  proceed  to  the  application  of  the  method  to  the 

development  of  algebraic  expressions  in  series. 

z    * 
Ex.  1.  Let  it  be  proposed,  to  develop  the  expression  — r—  in 

X  — T~  z 

series,  according  to  the  ascending  powers  of  x.     In  order  to  this, 
we  assume 

-4— =  A  +  Ba;  +  Ca:2   ,   D3^   ,   E^4  j_  ^  ^  ^ 
z-^x  '  '  '  '  ' 

the  coefficients  A,  B,  C,  being  independent  of  x. 

Freeing  the  first  member  from  its  denominator,  transposing 

and  arranging  with  reference  to  ar,  we  obtain 


UNDETERMINED   COEFFICIENTS. 


329 


—  2\  A  B  C| 

we  have,  therefore,  the  series  of  equations 
Az  —  z  =  0,  Bz-}-A  =  0,  Cz+B  =  0,  Dz  +  C=0,  &c., 
from  which  to  deduce  the  values  of  A,  B,  C,  &c.     Performing 
the  operations  and  substituting  in  the  assumed  expression,  we 
obtain  for  the  development  required 

Ex.  2.   Expand  the   fraction —       ^  into  an  infinitis 

X  ~~~  <fe  X  — r"  X 

series. 

Assume 
fee. 


Z        X 

Z  -\-  X  z 


1 


l^2x  +  s^ 


=:A4-Ba:  +  Ca:2  +  D^3  +  Ea:*  + 


Freeing  from  denominator,  transposing  and  reducing 

1  =  A+    B    x-\-    Q.    2:2+    D    ^_j.    E     a;*  +  &c. 
—  2A 


xJr    C 

2:2+     D 

2:^+    E 

—  2B 

-2C 

—  2D 

A 

B 

G 

from   which  we    obtain    A  =  1,  B 
1 


whence 


\^2x-\-7? 


2,  C==3,  D  =  4,  &c., 
=  l+2a;  +  3rc2  +  42:»  +  52r5+&c. 


1  +22: 
Ex.  3.  Expand  the  fraction  - — -5- -^  into  an  infinite  series. 

Ans.  l+32:  +  42:2  +  72:^+ll2;^  +  182:«  +  292;«+,&c. 

1  +  22; 
Ex.  4.  Expand         ^    into  an  infinite  series. 

1  O  X 

Ans.  l  +  52:  +  152:2+452.-3  +  1352:^+,  &c. 
•   What  is  the  law  of  the  coefficients  in  the  three  preceding 
series  ? 

Ex.  5.  Develop  (a  —  xY  in  series. 

Assume  (a  — 2:)*=A  +  Ba:  +  C2r»  +  Da:»  +  &c. 


Squaring,  (a— 2;)=  A^  +  2  A  B  2;  +       B^ 

2AC 


2  2:2  +  2BC|2^+&c.. 
2AD 


whence  A  =  ai  ,  B 


1 
2ai 


C  = 


2.4  aJ 


&c. 


from  which  the  development  sought  will  be  easily  obtained* 


330  ELEMENTS   OF   ALGEBRA. 

265w  Let  it  be  required  next  to  expand  ^ —^  into  an  infi- 

nite  series.  * 

If  we  assume ^  =  A-f-Ba;  +  Ca;2  4-Dr'-j->  &c.,  and 

determine  the  coefficients  accordingly,  we  shall  have  —  1  =  0, 
3  A=0,  3  B  —  A  =  0,  &c.,  the  first  of  which  is  absurd. 

The  proposed  cannot,  therefore,  be  developed  in  this  form. 
Indeed,  if  we  put  in  the  proposed  fraction  a;  =  0,  the  fraction 
takes  the  form  of  infinity.  The  development,  therefore,  for  this 
hypothesis,  should  take  the  same  form.  But  in  order  to  this  it 
must  contain,  it  is  obvious,  a  term  in  x  with  a  negative  exponent. 
"We  assume,  therefore, 

from  which  we  obtain  successively 

It  is  usual  to  assume  the  development  so  that  it  shall  proceed 
according  to  the  ascending  powers  of  x,  beginning  with  a:".  But 
this  form  will  not  always,  apply.  And  in  any  case  in  which  it 
is  not  applicable  the  fact  will  become  evident,  as  in  the  preced- 
ing example,  by  the  appearance  of  some  absurdity  in  the  result 
of  the  operation. 

The  form  which  the  development  should  take  may,  in 
general,  be  discovered  at  the  outset,  by  putting  a;  =  0  in  the 
function  to  be  developed,  and  observing  the  result.  If  the  func- 
tion, on  the  hypothesis  a;  =  0,  is  finite,  the  development  should 
be  taken  according  to  the  ascending  powers  of  x,  beginning  with 
Qp.  If  the  function  on  this  hypothesis  becomes  0,  the  first  term 
of  the  development  should  contain  x.  If  it  takes  the  form  of 
infinity,  the  first  term  of  the  development  should  contain  x  with 
a  negative  exponent. 

266.  The  following  miscellaneous  examples  will  serve  as  an 
additional  exercise. 


THE    DIFFERENTIAL   METHOD.  331 


Ex.  1.  To  develop  y-* —  in  series. 

Ans.  l  +  2a:  +  2a:2  +  2a:3+,  &c. 

Ex.   2.  Develop  -^ — ; — r^  in  series. 
(l+a:)2 

Ans.  1  — 2a;  +  3a;2_42;3  +  5a:*— ,  &c. 

Ex.  3.  Expand  (1  —  x)^  into  an  infinite  series. 

^^'-  ^-2"24~2A6-2A6:8^'^'- 

3  ^2 J 

Ex.  4.  Decompose  — - — —  into  partial  fractions. 

X  [x  ~\-  i)  [x  —  1) 

Ans. 


a;   '   a:-j-  1       x —  1 
The  Differential  Method. 

267.  Let  there  be  any  series  represented  by  a,  b,  c,  d,  &c. ; 
if  we  subtract  the  first  term  from  the  second,  the  second  from 
the  third,  and  so  on,  the  differences  thus  obtained  will  form  a 
new  series  called  the  Jirst  order  of  differences.  If  we  subtract 
again,  the  first  term  from  the  second,  the  second  from  the  third, 
&c.,  in  this  last  series,  the  differences  thus  obtained  will  form  a 
third  series,  called  the  second  order  of  differences,  and  so  on. 

Thus,  the  series  of  square  numbers,  with  the  several  orders 
of  differences,  is  as  follows : 

1        4        9        16        25        36        49 
3        5        7  9  11         13         1st  DifTerence. 

2        2         2  2  2  2d  " 

0        0  0  0  3d 

If  in  a  proposed  series  the  first  differences  are  all  the  same, 
or  constant,  the  series  is  called  a  difference  series  of  the  first 
order.  If  the  second  differences  are  constant,  it  is  called  a 
difference  series  of  the  second  order,  and  so  on.  Thus,  the 
series  of  square  numbers  above  is  a  difference  series  of  the 
second  order. 

Ex.  Of  what  order  is  the  series  1,  4,  10,  20,  35,  56 1 


332  ELEMENTS    OF    ALGEBRA. 

268.  From  wiiat  has  been  done  we  shall  have  the  following 
questions  to  solve  in  respect  to  series :  1°.  To  find  the  succes- 
sive differences  of  the  terms  of  a  series.  2°.  By  means  of 
these  to  find  any  intermediate  term,  and  the  sum  of  all  the 
terms. 

1.  Resuming  the  general  series,  we  have 

a,  b,  c,  df         e,  &c. 

3—  a,        c  —  b,        d  —  c,        e  —  d  1st  DifT. 

c  —  2b-{-a,d  —  2c-\-       b,        e  —  2d-{-c      2d      " 

^__3c-}-3i  — a,  &c.,  3d      " 

<( 

If  we  now  represent  the  first  terms  of  the  successive  orders 
of  differences  by  Di,  D2,  &c.,  we  shall  have,  reversing  the  order 
of  the  terms, 

Di  =  — «+    b 

D2=      a  —  2b-^    c 

D3  =  — a  +  33  — 3c  +  d: 

D4==    &c. 
In  which  it  will  be  perceived  that  the  coefficients  of  the  several 
terms  correspond  with  those  in  the  expansion  of  a  binomiaL 
And  we  shall  have,  generally, 

D.  =  ±a^^7ib±n-j^  c^^n   ^ ^ ^(^zt&c., 

the  upper  sign  corresponding  to  the  case  in  which  the  difference 
is  even,  and  the  lower  to  the  case  in  which  it  is  odd. 

By  means  of  this  formula,  we  readily  find  the  first  term  in 
&ny  order  of  differences. 

Ex.  1.  What  is  the  first  term  of  the  third  order  of  differences 
in  the  series  of  cubes  1,  8,  27,  64,  125,  &c.  ? 

In  this  case,  n  being  odd,  we  use  the  lower  signs  of  the  for- 
mula, and  we  have 

D3  =  — 1+3.8  — 3.27  +  64  =  6. 
,     Ex.  2.  What  is  *  the  first  term  of  the  fourth  order  of  differ- 
ences in  the  series  7,  12,  21,  36,  62  ?  .  Ans.  3. 


THE    DIFFERENTIAL   METHOD.  3^ 

Px.  3.  What  is  the  first  term  of  the  fifth  order  of  differences 
in  the  series  1,  6,  21,  56,  126,  252,  &c.  ?  Ans.  1. 

2.  Let  it  be  required  next  to  find  any  intermediate  term  of 
the  series. 

From  the  expressions  Dj,  D2,  D3,  &c.,  we  obtain 

c  =  a-f2Di+    D2 

ef  =  a -f- 3  Di +  3  D2  +  D3, 

in  which  it  will  be  perceived  that  the  coefficients  of  the  succes- 
sive terms  are  the  coefficients  of  the  power  of  a  binomial  one 
degree  less  than  the  number  of  the  term ;  whence,  putting  L 
for  the  72th  term  of  the  series 

The  series  will  terminate  if  the  diflferences  become  0  after  a 
certain  order.  Otherwise  it  will  be  infinite,  and  L  can  be  deter- 
mined only  approximately. 

Ex.  1.  What  is  the  50th  term  of  the  series  1,  4,  8,  13,  &c.  ? 

Here  a  =  1,  Di  ==  3,  D2  =  1,  D3  =  0.  Ans.  1324. 

Ex.  2.  Required  the  tenth  term  of  the  series  1,  4,  8,  13,  19, 
&c.  Ans.  64. 

Ex.  3.  Required  the  twentieth  term  of  the  series  1,  5, 15, 35, 
70,  126,  &c.  Ans.  8855. 

Ex.  4.  What  is  the  twelfth  term  of  the  order  of  cubes,  or,  in 
other  words,  whafls  the  cube  of  12?  Ans.  1728. 

3.  Let  it  be  required  next  to  find  the  sum  of  any  number  n 
terms  of  the  series 

a,  ^,  c,  dt  e,  &c. 
Assume  the  series 

^,  a^  a  -\-  bt  a  -\-  b  -\'  Ct  a  -\-  h  -\-  c  '\-  d^  &c. 
Subtracting  each  term  from  the  next  succeeding,  we  obtain  a, 
b,  c,  <?,  &c.,  the  proposed  series.  From  this  it  follows  that  the 
sum  of  n  terms  of  the  proposed  series  is  the  (?t  -|-  l)th  term  of 
the  assumed  series,  and  that  the  wth  order  of  differences  in  the 
first  series  is  the  same  as  the  (n-(-  l)th  order  in  the  second. 


334  ELEMENTS   OF   ALGEBRA. 

We  shall  have,  therefore,  by  substituting  in  the  formula  for  L 

above, 

0  for  a,n-\-\  for  n,  a  for  Di,  Dj  for  D2,  &c.,  .  .  . 

71^1               {n—\){n  —  2) 
S  =  7za  +  7i-^— Di  +  ?i^^ -^A iD2  +  ... 

which  is  the  expression  for  the  sum  of  any  number  n  tearms  of 
the  series. 

Ex.  1.  Required  the  sum  of  12  terms  of  the  series 
1,  4,  8,  13,  19 

Here  a  =  1,  Di  =  3,  D2  =  1,  D3  =  0,  %  =  12.     Ans.  430. 

Ex.  2.  Required  the  sum  of  n  terms  of  the  series  1,  2,  3,  4, 

5,  &c.  .        11? -\-n 

Ans.  -^. 

Ex.  3.  Required  the  sum  of  ten  terms  of  the  series  1,  5,  15, 
35,  70,  126,  &c.  Ans.  2002. 

Interpolation. 

269.  The  formula  for  L  may  be  applied  to  the  process  of 
interpolation,  or  that  of  finding  numbers  intermediate  between 
those  contained  in  tables. 

Suppose  that  we  have  a  table  of  the  square  roots  of  numbers, 
from  1  to  100,  and  that  we  wish,  for  example,  to  insert  the 
square  roots  of  the  intermediate  numbers  to  every  ^  of  a  unit. 
If  the  first  differences  of  the  roots  of  the  numbers  already  in  the 
tables  were  constant,  we  should  have  merely  to  find  proportional 
parts  of  the  difference  between  any  two  consecutive  roots,  which 
we  should  add  to  the  roots  respectively.  This  would  be 
the  method  by  first  differences.  But  the  first  differences  not 
being  constant,  the  second  must  be  taken  into  consideration, 
and  so  on,  until  the  diflferences  become  constant,  or  the  requisite 
degree  of  approximation  has  been  obtained.  For  this  purpose 
we  employ  the  formula  L  above.  Putting,  for  convenience,  in 
this  formula  w,  to  represent  the  distance  of  any  term  from  the 
first,  we  have  x  =  n  —  1,  and  the  formula  becomes 


SUMMATION    OF   INFINITE    SERIES. 


335 


L  =  a  +  a:  Di  -f- 


x{x—l) 


^(^-l)(^-2) 


Let  three  contiguous  numbers  of  the  table,  with  the  square 
roots  and  differences  to  D2,  be  as  follows ;  it  is  required  to  find 
the  square  root  of  94^,  or  94.25. 


No. 

Square  root. 

D: 

D. 

94 
95 
76 

9.69536 
9.74679 
9.79796 

.05143 
.05117 

-  .00026 

In  the  present  case  we  shall  have  a  =  9.69536,  Dj  =  .05143, 
D2  =  — .00026,  anda;  =  :^;  whence 

L==  9.69536  +  ^  (.05143) +  ^3_  (.00026) 

=  9.70824,     Ans. 
Ex.  2.  Given  the  logarithms  of  6,  7,  8,  9,  and  10,  to  find 
that  of  6.5. 


No. 

Logs. 

Di   1    D, 

D3 

D4 

6 

7 

8 

9 

10 

0.778151 
0.845098 
0.903090 
0.954243 
1.000000 

.066947 
.057992 
.051153 
.045757 

—  .008955 

—  .006839 

—  .005396 

.002116 
.001443 

-  000673 

L  =  a  4-  i  Di  —  ^  D2  +  -iV  D3  —  xf  ^  D4  =  0.812901  Ans. 

Ex.  3.  Given  the  natural  sine  of  37°  equal  to  .60182 

"        "        38°      "  .61566 

"        "        39°      "    '  .62932 

"        "        40°      "  .64279 

to  find  that  of  37°  30'.  Ans.  .60876. 

Summation  of  Infinite  Series. 

270.  'The  summation  of  an  infinite  series  consists  in  finding 
a  finite  expression  equivalent  to  the  series. 

Since  the  sum  of  a  series  must  evidently  depend  upon  the 
law  of  the  series,  no  formula  can  be  given  for  the  summation  of 
series  which  will  apply  universally.  A  great  variety  of  useful 
series  may  be  summed  by  the  following  principles. 


336 


ELEMENTS   OF   ALGEBRA. 


Let  there  be  a  series,  the  terms  of  which  shall  be  represented 
by  the  fraction, 

pq  q  q 


whence 


n{n-{-p)      '71      n-\-p 

p  \n       n-^-pJ 


n{n-{-p)      p 
If,  then,  a  series  be  represented  by 


,  its  sum,  it  IS 


n  {7i-\-p) 
evident,  will  be  equal  to  a  pth  part  of  the  difference  of  the  two 

series  represented  by  -  and  — ^ — . 
^  *'  w         n-{-p 

Ex.  1.  What  is  the  sum  of  n  terms  of  the  series 

1_        J_        i_        i_  2         ' 

1.2'    2.3'    3.4'    4.5  *  *  '  * 
Here,  in  the  general  term  g'=  1,  ^  =  1,  ?i5=  1,  2,  3,  &c., 
whence  we  have  for  the  sum  S, 


S  =  l+i  +  i+...i 


2 
—  1 

2 


«  +  l' 


Ans. 


n      71+ 1 
If  71  =  00,  the  sum,  it  is  evident,  will  be  1. 
Ex.  2.  What  is  the  sum  of  n  terms  of  the  series 

1.3 
Here  q==:l,p  =  2,  and  7i=  1,  3,  5,  7  .  .  . ;  whence 


3:5'     577'     7:9'     9li- -to  infinity. 


S  = 


^+1+1+1 


1 


2n—l 
1 


271-1  271+1 


-,  Ans. 


271+r 

If  w  =  00,  the  sum,  it  is  evident,  will  be  J. 


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